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5772:
6959:. Several alternative approaches have been developed to eliminate such nuisance parameters, so that a likelihood can be written as a function of only the parameter (or parameters) of interest: the main approaches are profile, conditional, and marginal likelihoods. These approaches are also useful when a high-dimensional likelihood surface needs to be reduced to one or two parameters of interest in order to allow a
15453:
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47:
3683:
2951:
5514:
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10521:{\displaystyle {\begin{aligned}&{\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \mid x_{1},\ldots ,x_{n})}{\partial \beta }}\\={}&{\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \mid x_{1})}{\partial \beta }}+\cdots +{\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \mid x_{n})}{\partial \beta }}={\frac {n\alpha }{\beta }}-\sum _{i=1}^{n}x_{i}.\end{aligned}}}
7691:
3529:
5222:
2926:
7490:
9329:
6023:
4101:{\displaystyle {\begin{aligned}&\mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x_{j})=\mathop {\operatorname {arg\,max} } _{\theta }\left)\right]\\={}&\mathop {\operatorname {arg\,max} } _{\theta }\left=\mathop {\operatorname {arg\,max} } _{\theta }f(x_{j}\mid \theta ).\end{aligned}}}
3266:{\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}{\mathcal {L}}(\theta \mid x\in )=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\Pr(x_{j}\leq x\leq x_{j}+h\mid \theta )=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx,}
9453:
7522:
5767:{\displaystyle \left|{\frac {\partial f}{\partial \theta _{r}}}\right|<F_{r}(x)\,,\quad \left|{\frac {\partial ^{2}f}{\partial \theta _{r}\,\partial \theta _{s}}}\right|<F_{rs}(x)\,,\quad \left|{\frac {\partial ^{3}f}{\partial \theta _{r}\,\partial \theta _{s}\,\partial \theta _{t}}}\right|<H_{rst}(x)}
4251:
10884:
of the data given the parameter (since the parameter is then a random variable) and (ii) a measure or amount of information brought by the data about the parameter value or even the model. Due to the introduction of a probability structure on the parameter space or on the collection of models, it is
10757:
n 1922, I proposed the term 'likelihood,' in view of the fact that, with respect to , it is not a probability, and does not obey the laws of probability, while at the same time it bears to the problem of rational choice among the possible values of a relation similar to that which probability bears
6971:
It is possible to reduce the dimensions by concentrating the likelihood function for a subset of parameters by expressing the nuisance parameters as functions of the parameters of interest and replacing them in the likelihood function. In general, for a likelihood function depending on the parameter
4455:
is always one. Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with in the manner shown above.
4327:
of the probability distribution relative to a common dominating measure. The likelihood function is this density interpreted as a function of the parameter, rather than the random variable. Thus, we can construct a likelihood function for any distribution, whether discrete, continuous, a mixture, or
11109:
Each independent sample's maximum likelihood estimate is a separate estimate of the "true" parameter set describing the population sampled. Successive estimates from many independent samples will cluster together with the population's "true" set of parameter values hidden somewhere in their midst.
8245:
is the sum of the log-probability of the individual events. In addition to the mathematical convenience from this, the adding process of log-likelihood has an intuitive interpretation, as often expressed as "support" from the data. When the parameters are estimated using the log-likelihood for the
6079:
The above conditions are sufficient, but not necessary. That is, a model that does not meet these regularity conditions may or may not have a maximum likelihood estimator of the properties mentioned above. Further, in case of non-independently or non-identically distributed observations additional
4583:
The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the
11140:
As more data are observed, instead of being used to make independent estimates, they can be combined with the previous samples to make a single combined sample, and that large sample may be used for a new maximum likelihood estimate. As the size of the combined sample increases, the size of the
11136:
converts the region's log-likelihood differences into the "confidence" that the population's "true" parameter set lies inside. The art of choosing the fixed log-likelihood difference is to make the confidence acceptably high while keeping the region acceptably small (narrow range of estimates).
10766:
I stress this because in spite of the emphasis that I have always laid upon the difference between probability and likelihood there is still a tendency to treat likelihood as though it were a sort of probability. The first result is thus that there are two different measures of rational belief
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7805:
11141:
likelihood region with the same confidence shrinks. Eventually, either the size of the confidence region is very nearly a single point, or the entire population has been sampled; in both cases, the estimated parameter set is essentially the same as the population parameter set.
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8409:
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quantifies the heuristic rule by showing that the difference in the logarithm of the likelihood generated by the estimate's parameter values and the logarithm of the likelihood generated by population's "true" (but unknown) parameter values is asymptotically
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9338:
10767:
appropriate to different cases. Knowing the population we can express our incomplete knowledge of, or expectation of, the sample in terms of probability; knowing the sample we can express our incomplete knowledge of the population in terms of likelihood.
5165:
and asymptotic normality of the maximum likelihood estimator, additional assumptions are made about the probability densities that form the basis of a particular likelihood function. These conditions were first established by Chanda. In particular, for
4649:
parameter space for the maximum likelihood estimator to exist. While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values might be unknown. In that case,
7339:
7132:
6749:
5394:{\displaystyle {\frac {\partial \log f}{\partial \theta _{r}}}\,,\quad {\frac {\partial ^{2}\log f}{\partial \theta _{r}\partial \theta _{s}}}\,,\quad {\frac {\partial ^{3}\log f}{\partial \theta _{r}\,\partial \theta _{s}\,\partial \theta _{t}}}\,}
4113:
8105:
4908:
4636:
in various proofs involving likelihood functions, and need to be verified in each particular application. For maximum likelihood estimation, the existence of a global maximum of the likelihood function is of the utmost importance. By the
7054:
7957:
Sometimes we can remove the nuisance parameters by considering a likelihood based on only part of the information in the data, for example by using the set of ranks rather than the numerical values. Another example occurs in linear
6544:
5877:
7686:{\displaystyle {\hat {\beta }}_{1}=\left(\mathbf {X} _{1}^{\mathsf {T}}\left(\mathbf {I} -\mathbf {P} _{2}\right)\mathbf {X} _{1}\right)^{-1}\mathbf {X} _{1}^{\mathsf {T}}\left(\mathbf {I} -\mathbf {P} _{2}\right)\mathbf {y} }
6128:
1079:
8250:, each data point is used by being added to the total log-likelihood. As the data can be viewed as an evidence that support the estimated parameters, this process can be interpreted as "support from independent evidence
8262:, the support (log-likelihood) of a model, given an event, is the negative of the surprisal of the event, given the model: a model is supported by an event to the extent that the event is unsurprising, given the model.
4343:
The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses
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to the problem of predicting events in games of chance. . . . Whereas, however, in relation to psychological judgment, likelihood has some resemblance to probability, the two concepts are wholly distinct. . . ."
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7696:
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8903:
3524:{\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x\in )=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx.}
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of the population that the observed sample was drawn from. Heuristically, it makes sense that a good choice of parameters is those which render the sample actually observed the maximum possible
10242:
8709:
8413:
Just as the likelihood, given no event, being 1, the log-likelihood, given no event, is 0, which corresponds to the value of the empty sum: without any data, there is no support for any models.
6070:
6658:
8241:
Given the independence of each event, the overall log-likelihood of intersection equals the sum of the log-likelihoods of the individual events. This is analogous to the fact that the overall
8967:
6295:. Numerous other tests can be viewed as likelihood-ratio tests or approximations thereof. The asymptotic distribution of the log-likelihood ratio, considered as a test statistic, is given by
561:
11080:. The value of the likelihood serves as a figure of merit for the choice used for the parameters, and the parameter set with maximum likelihood is the best choice, given the data available.
2921:{\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x\in )=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}{\mathcal {L}}(\theta \mid x\in ),}
11657:
Kass, Robert E.; Tierney, Luke; Kadane, Joseph B. (1990). "The
Validity of Posterior Expansions Based on Laplace's Method". In Geisser, S.; Hodges, J. S.; Press, S. J.; Zellner, A. (eds.).
7182:
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In many cases, the likelihood is a function of more than one parameter but interest focuses on the estimation of only one, or at most a few of them, with the others being considered as
1372:
475:
7996:
9714:
4762:
4754:
7485:{\textstyle \beta _{2}(\beta _{1})=\left(\mathbf {X} _{2}^{\mathsf {T}}\mathbf {X} _{2}\right)^{-1}\mathbf {X} _{2}^{\mathsf {T}}\left(\mathbf {y} -\mathbf {X} _{1}\beta _{1}\right)}
9637:
9324:{\displaystyle p(x\mid {\boldsymbol {\theta }})=h(x)\exp {\Big (}\langle {\boldsymbol {\eta }}({\boldsymbol {\theta }}),\mathbf {T} (x)\rangle -A({\boldsymbol {\theta }}){\Big )}.}
5457:
5218:
11083:
The specific calculation of the likelihood is the probability that the observed sample would be assigned, assuming that the model chosen and the values of the several parameters
8210:
functions, maximizing the likelihood is equivalent to maximizing the log-likelihood. But for practical purposes it is more convenient to work with the log-likelihood function in
6018:{\displaystyle \mathbf {I} (\theta )=\int _{-\infty }^{\infty }{\frac {\partial \log f}{\partial \theta _{r}}}\ {\frac {\partial \log f}{\partial \theta _{s}}}\ f\ \mathrm {d} z}
5509:
4940:
5068:
7059:
10633:
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9192:
4719:
1928:
10232:, then the joint log-likelihood will be the sum of individual log-likelihoods, and the derivative of this sum will be a sum of derivatives of each individual log-likelihood:
9675:
7307:
6992:
9123:
8738:
8107:
This follows from the definition of independence in probability: the probabilities of two independent events happening, given a model, is the product of the probabilities.
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3308:
1995:
1761:
1648:
1491:
1407:
759:
640:
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8204:
7974:
A partial likelihood is an adaption of the full likelihood such that only a part of the parameters (the parameters of interest) occur in it. It is a key component of the
4378:
2447:
2183:
1959:
9448:{\displaystyle \ell ({\boldsymbol {\theta }}\mid x)=\langle {\boldsymbol {\eta }}({\boldsymbol {\theta }}),\mathbf {T} (x)\rangle -A({\boldsymbol {\theta }})+\log h(x).}
8835:
8650:
7242:
2159:
1611:
1584:
829:
794:
5148:
8805:
7906:
7872:
7517:
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11128:. The region surrounds the maximum-likelihood estimate, and all points (parameter sets) within that region differ at most in log-likelihood by some fixed value. The
8172:
6380:
6349:
4632:
In the context of parameter estimation, the likelihood function is usually assumed to obey certain conditions, known as regularity conditions. These conditions are
992:
9760:
8923:
8766:
8111:
4679:
4622:
4246:{\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x_{j})=\mathop {\operatorname {arg\,max} } _{\theta }f(x_{j}\mid \theta ),}
2487:
2467:
2394:
2370:
2282:
1654:
1427:
1333:
1289:
1208:
1101:
985:
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849:
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422:
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9915:
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2630:
10749:, in two research papers published in 1921 and 1922. The 1921 paper introduced what is today called a "likelihood interval"; the 1922 paper introduced the term "
4305:
4278:
2552:
9517:
6404:
4328:
otherwise. (Likelihoods are comparable, e.g. for parameter estimation, only if they are RadonâNikodym derivatives with respect to the same dominating measure.)
8121:
The empty product has value 1, which corresponds to the likelihood, given no event, being 1: before any data, the likelihood is always 1. This is similar to a
6997:
2513:
2423:
906:
10981:
10961:
10939:
10919:
9935:
6825:
5792:
5481:
5187:
4602:
4578:
4558:
4538:
4453:
4433:
2946:
2262:
2242:
2211:
1447:
1309:
1269:
1188:
1168:
1148:
1128:
965:
938:
720:
660:
517:
497:
2604:
2075:
1768:
6935:
will be the same as a 95% confidence interval (19/20 coverage probability). In a slightly different formulation suited to the use of log-likelihoods (see
5797:
10817:. For each of the proposed foundations, the interpretation of likelihood is different. The four interpretations are described in the subsections below.
5073:
11849:
2002:
1662:
10555:
6577:
Since the actual value of the likelihood function depends on the sample, it is often convenient to work with a standardized measure. Suppose that the
7847:
Since graphically the procedure of concentration is equivalent to slicing the likelihood surface along the ridge of values of the nuisance parameter
12368:
Rao, B. Raja (1960). "A formula for the curvature of the likelihood surface of a sample drawn from a distribution admitting sufficient statistics".
10781:(1972) established the axiomatic basis for use of the log-likelihood ratio as a measure of relative support for one hypothesis against another. The
6083:
In
Bayesian statistics, almost identical regularity conditions are imposed on the likelihood function in order to proof asymptotic normality of the
4459:
2289:
12287:
14550:
350:
that (presumably) generated the observations. When evaluated on the actual data points, it becomes a function solely of the model parameters.
15055:
309:
7935:
One example occurs in 2Ă2 tables, where conditioning on all four marginal totals leads to a conditional likelihood based on the non-central
7800:{\textstyle \mathbf {P} _{2}=\mathbf {X} _{2}\left(\mathbf {X} _{2}^{\mathsf {T}}\mathbf {X} _{2}\right)^{-1}\mathbf {X} _{2}^{\mathsf {T}}}
8579:, it is easier to compute the stationary points of the log-likelihood of independent events than for the likelihood of independent events.
6939:), the test statistic is twice the difference in log-likelihoods and the probability distribution of the test statistic is approximately a
6548:
The likelihood ratio is not directly used in AIC-based statistics. Instead, what is used is the relative likelihood of models (see below).
6264:
states that degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio.
10638:
8498:
7932:
for the nuisance parameters, and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters.
15205:
12314:
Tarone, Robert E.; Gruenhage, Gary (1975). "A Note on the
Uniqueness of Roots of the Likelihood Equations for Vector-Valued Parameters".
10880:), given specified data or other evidence, the likelihood function remains the same entity, with the additional interpretations of (i) a
9036:
100:
14829:
13470:
6245:{\displaystyle \Lambda (\theta _{1}:\theta _{2}\mid x)={\frac {{\mathcal {L}}(\theta _{1}\mid x)}{{\mathcal {L}}(\theta _{2}\mid x)}}.}
2218:
12341:
Rai, Kamta; Van Ryzin, John (1982). "A Note on a
Multivariate Version of Rolle's Theorem and Uniqueness of Maximum Likelihood Roots".
9155:. The probability distribution function (and thus likelihood function) for exponential families contain products of factors involving
522:
9333:
Each of these terms has an interpretation, but simply switching from probability to likelihood and taking logarithms yields the sum:
8404:{\displaystyle \log {\frac {{\mathcal {L}}(A)}{{\mathcal {L}}(B)}}=\log {\mathcal {L}}(A)-\log {\mathcal {L}}(B)=\ell (A)-\ell (B).}
566:
14603:
12100:
4945:
1555:) for the probability of a coin landing heads-up (without prior knowledge of the coin's fairness), given that we have observed HHT.
182:
9606:{\displaystyle \ell ({\boldsymbol {\eta }}\mid x)=\langle {\boldsymbol {\eta }},\mathbf {T} (x)\rangle -A({\boldsymbol {\eta }}).}
1493:) for the probability of a coin landing heads-up (without prior knowledge of the coin's fairness), given that we have observed HH.
15042:
8589:
6556:
6111:
5880:
4456:
For an observation from the discrete component, the likelihood function for an observation from the discrete component is simply
9460:
2161:, we can calculate the corresponding likelihood. The result of such calculations is displayed in Figure 1. The integral of
436:
10059:{\displaystyle \log {\mathcal {L}}(\alpha ,\beta \mid x)=\alpha \log \beta -\log \Gamma (\alpha )+(\alpha -1)\log x-\beta x.\,}
8840:
6110:
This section is about the likelihood ratio in general. For the use of likelihood ratios in interpreting diagnostic tests, see
13110:
13057:
12926:
12054:
11996:
11967:
11938:
11330:
1653:
Imagine flipping a fair coin twice, and observing two heads in two tosses ("HH"). Assuming that each successive coin flip is
6936:
4316:
13465:
13165:
11506:
Mascarenhas, W.F. (2011). "A mountain pass lemma and its implications regarding the uniqueness of constrained minimizers".
10881:
9888:{\displaystyle {\mathcal {L}}(\alpha ,\beta \mid x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}.}
11110:
The difference in the logarithms of the maximum likelihood and adjacent parameter sets' likelihoods may be used to draw a
10177:{\displaystyle {\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \mid x)}{\partial \beta }}={\frac {\alpha }{\beta }}-x.}
8972:
1586:
that expresses the "fairness" of the coin. The parameter is the probability that a coin lands heads up ("H") when tossed.
14069:
13217:
8575:
is zero); since the derivative of a sum is just the sum of the derivatives, but the derivative of a product requires the
7841:
10885:
possible that a parameter value or a statistical model have a large likelihood value for given data, and yet have a low
8658:
6032:
11465:"On the existence and uniqueness of the maximum likelihood estimate of a vector-valued parameter in fixed-size samples"
6916:
in
Bayesian statistics. Likelihood intervals are interpreted directly in terms of relative likelihood, not in terms of
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14744:
13080:
13038:
12978:
12952:
12899:
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12417:
12269:
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12082:
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11752:
11666:
11587:
11447:
11380:
11355:
11305:
11041:
11019:
9718:. Thus for example the maximum likelihood estimate can be computed by taking derivatives of the sufficient statistic
8928:
7987:
302:
265:
11058:
that summarizes a single sample from a population, whose calculated value depends on a choice of several parameters
11012:
15457:
15030:
14904:
9159:. The logarithm of such a function is a sum of products, again easier to differentiate than the original function.
8223:
7148:
6931:
is a single real parameter, then under certain conditions, a 14.65% likelihood interval (about 1:7 likelihood) for
2635:
192:
8442:
6754:
1504:
15088:
14749:
14494:
13865:
13455:
12962:
12098:
Venzon, D. J.; Moolgavkar, S. H. (1988). "A Method for
Computing Profile-Likelihood-Based Confidence Intervals".
7978:: using a restriction on the hazard function, the likelihood does not contain the shape of the hazard over time.
3671:{\displaystyle \lim _{h\to 0^{+}}{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx=f(x_{j}\mid \theta ).}
3534:
1830:
425:
218:
95:
10530:
To complete the maximization procedure for the joint log-likelihood, the equation is set to zero and solved for
15139:
14351:
14158:
14047:
14005:
12944:
12261:
8456:, i.e. whether or not the data "support" one hypothesis (or parameter value) being tested more than any other.
8453:
1338:
343:
156:
17:
14079:
9688:
15382:
14341:
13244:
8468:
8247:
8211:
4724:
354:
9162:
An exponential family is one whose probability density function is of the form (for some functions, writing
15506:
14933:
14882:
14867:
14857:
14726:
14598:
14565:
14391:
14346:
14176:
12990:"Efficiency Testing of Prediction Markets: Martingale Approach, Likelihood Ratio and Bayes Factor Analysis"
11242:
11185:
10814:
10750:
9620:
7127:{\textstyle \mathbf {\hat {\theta }} _{2}=\mathbf {\hat {\theta }} _{2}\left(\mathbf {\theta } _{1}\right)}
6744:{\displaystyle R(\theta )={\frac {{\mathcal {L}}(\theta \mid x)}{{\mathcal {L}}({\hat {\theta }}\mid x)}}.}
4320:
2524:
2222:
295:
187:
125:
11630:
Chen, Chan-Fu (1985). "On
Asymptotic Normality of Limiting Density Functions with Bayesian Implications".
11165:
10840:
8118:. In such a situation, the likelihood function factors into a product of individual likelihood functions.
15445:
15277:
15078:
15002:
14303:
14057:
13726:
13190:
10771:
Fisher's invention of statistical likelihood was in reaction against an earlier form of reasoning called
7963:
7936:
7917:
6578:
5404:
5192:
6947:
likelihood interval is the same as the 0.954 confidence interval; assuming difference in df's to be 1).
5486:
4917:
15471:
15162:
15134:
15129:
14877:
14636:
14542:
14522:
14430:
14141:
13959:
13442:
13314:
13102:
12574:
12402:
11790:
9523:, so in these coordinates, the log-likelihood of an exponential family is given by the simple formula:
7990:
7975:
5045:
177:
146:
11273:
14894:
14662:
14383:
14308:
14237:
14166:
14086:
14074:
13944:
13932:
13925:
13633:
13354:
11684:
11603:
Heyde, C. C.; Johnstone, I. M. (1979). "On
Asymptotic Posterior Normality for Stochastic Processes".
10798:
10609:
10189:
9649:
9165:
8467:(the curvature of the log-likelihood). Thus, the graph has a direct interpretation in the context of
8235:
8115:
4691:
4324:
1891:
945:
335:
239:
120:
10762:
The concept of likelihood should not be confused with probability as mentioned by Sir Ronald Fisher
7250:
6975:
6943:
with degrees-of-freedom (df) equal to the difference in df's between the two models (therefore, the
6809:
whose relative likelihood is greater than or equal to a given threshold. In terms of percentages, a
15377:
15144:
15007:
14692:
14657:
14621:
14406:
13848:
13757:
13716:
13628:
13319:
13158:
12678:
12162:
11232:
11129:
11103:
11006:
8769:
8215:
7135:
6940:
6751:
Thus, the relative likelihood is the likelihood ratio (discussed above) with the fixed denominator
6280:
6255:
5154:
while informally appealing to a mountain pass property. Mascarenhas restates their proof using the
428:
260:
172:
12503:
11682:
Buse, A. (1982). "The
Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note".
11541:
Chanda, K.C. (1954). "A note on the consistency and maxima of the roots of likelihood equations".
9099:
8714:
8100:{\displaystyle \Lambda (A\mid X_{1}\land X_{2})=\Lambda (A\mid X_{1})\cdot \Lambda (A\mid X_{2}).}
7814:
6621:
6588:
4383:
3278:
1967:
1733:
1620:
1464:
1377:
729:
610:
15286:
14899:
14839:
14776:
14414:
14398:
14136:
13998:
13988:
13838:
13752:
12623:
Fienberg, Stephen E (1997). "Introduction to R.A. Fisher on inverse probability and likelihood".
11874:
11212:
10775:. His use of the term "likelihood" fixed the meaning of the term within mathematical statistics.
7962:, where considering a likelihood for the residuals only after fitting the fixed effects leads to
6552:
4903:{\displaystyle \mathbf {H} (\theta )\equiv \left_{i,j=1,1}^{n_{\mathrm {i} },n_{\mathrm {j} }}\;}
909:
723:
11433:
9615:
In words, the log-likelihood of an exponential family is inner product of the natural parameter
8185:
4347:
2428:
2164:
1933:
15324:
15254:
15047:
14984:
14739:
14626:
13623:
13520:
13427:
13306:
13205:
13026:
12480:(1921). "On the "probable error" of a coefficient of correlation deduced from a small sample".
12435:
12231:"Why we always put log() before the joint pdf when we use MLE (Maximum likelihood Estimation)?"
12019:
12013:
11090:
11023:
10742:
10732:
9680:
9130:
7940:
6894:
374:
151:
12044:
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11923:
11918:
11437:
8818:
8633:
7191:
6618:
values may be found by comparing the likelihoods of those other values with the likelihood of
2137:
1589:
1562:
799:
764:
15349:
15291:
15234:
15060:
14953:
14862:
14588:
14472:
14331:
14323:
14213:
14205:
14020:
13916:
13894:
13853:
13818:
13785:
13731:
13706:
13661:
13600:
13560:
13362:
13185:
13094:
12966:
12203:
11227:
10898:
10728:
9520:
8564:
8472:
8254:
and the log-likelihood is the "weight of evidence". Interpreting negative log-probability as
6921:
6276:
6268:
6115:
6088:
6084:
5155:
5131:
4638:
4335:, under which the probability density at any outcome equals the probability of that outcome.
2397:
1108:
431:(a more general definition is discussed below). Given a probability density or mass function
389:
339:
54:
8775:
7884:
7850:
7495:
7312:
15272:
14847:
14796:
14772:
14734:
14652:
14631:
14583:
14462:
14440:
14409:
14318:
14195:
14146:
14064:
14037:
13993:
13949:
13711:
13487:
13367:
12597:
12515:
12436:"On the history of maximum likelihood in relation to inverse probability and least squares"
12188:
11718:
11469:
11404:
11222:
11217:
9642:
9148:
8583:
8157:
7929:
6917:
6358:
6327:
5162:
234:
115:
85:
12547:
12285:
Foutz, Robert V. (1977). "On the Unique
Consistent Solution to the Likelihood Equations".
7049:{\textstyle \mathbf {\theta } =\left(\mathbf {\theta } _{1}:\mathbf {\theta } _{2}\right)}
4657:
More specifically, if the likelihood function is twice continuously differentiable on the
2527:
in specifying the likelihood function above is justified as follows. Given an observation
908:. Such an interpretation is a common error, with potentially disastrous consequences (see
8:
15501:
15419:
15344:
15267:
14948:
14712:
14705:
14667:
14575:
14555:
14527:
14260:
14126:
14121:
14111:
14103:
13882:
13772:
13762:
13671:
13450:
13406:
13324:
13249:
13151:
12762:
Introduction to Probability and Statistics from a Bayesian Viewpoint. Part 1: Probability
12625:
12440:
11396:
11207:
10877:
10772:
9745:
8908:
8751:
8422:
8255:
8207:
7952:
7913:
7878:
6960:
6909:
6905:
6572:
5039:
4664:
4642:
4607:
2472:
2452:
2379:
2355:
2267:
1412:
1318:
1274:
1193:
1086:
970:
865:
834:
685:
665:
407:
381:
66:
38:
12601:
12519:
10533:
10075:
9900:
9765:
2609:
2518:
15485:
15433:
15244:
15098:
14994:
14943:
14819:
14716:
14700:
14677:
14454:
14188:
14171:
14131:
14042:
13937:
13899:
13870:
13830:
13790:
13736:
13653:
13339:
13334:
13069:
12551:
12459:
12144:
12117:
11955:
11893:
11643:
11616:
11560:
11523:
11488:
11252:
11237:
11189:
10872:
given another random variable: for example the likelihood of a parameter value or of a
10865:
10069:
9739:
9142:
9126:
9090:
8492:
8464:
8460:
8231:
8219:
7185:
6956:
6389:
6303:
6292:
6073:
5884:
4283:
4256:
2530:
366:
283:
208:
110:
80:
9493:
8630:
In that sense, the maximum likelihood estimator is implicitly defined by the value at
15480:
15428:
15339:
15309:
15301:
15121:
15112:
15037:
14968:
14824:
14809:
14784:
14672:
14613:
14479:
14467:
14093:
14010:
13954:
13877:
13721:
13643:
13422:
13296:
13106:
13076:
13053:
13034:
12974:
12948:
12922:
12895:
12807:
12659:
12265:
12209:
12078:
12050:
12023:
11992:
11963:
11934:
11829:
11748:
11744:
11662:
11583:
11443:
11376:
11351:
11326:
11301:
11133:
11111:
11098:
11077:
10894:
10873:
10801:
should be. There are four main paradigms that have been proposed for the foundation:
8812:
8265:
A logarithm of a likelihood ratio is equal to the difference of the log-likelihoods:
7808:
7142:
6913:
6539:{\displaystyle O(A_{1}:A_{2}\mid B)=O(A_{1}:A_{2})\cdot \Lambda (A_{1}:A_{2}\mid B).}
6296:
6288:
6284:
6260:
6026:
4911:
2492:
2402:
2185:
over is 1/3; likelihoods need not integrate or sum to one over the parameter space.
1885:
885:
393:
331:
278:
213:
90:
62:
12909:
Boos, Dennis D.; Stefanski, L. A. (2013). "Likelihood Construction and Estimation".
12869:
Model Selection and Multimodel Inference: A practical information-theoretic approach
11889:
11527:
10789:, but were not adopted in a general treatment of the topic of statistical evidence.
5872:{\textstyle \,\int _{-\infty }^{\infty }H_{rst}(z)\mathrm {d} z\leq M<\infty \;.}
15364:
15319:
15083:
15070:
14963:
14938:
14872:
14804:
14682:
14290:
14183:
14116:
14029:
13976:
13795:
13666:
13460:
13344:
13259:
13226:
13132:
13011:
13001:
12936:
12914:
12830:
12799:
12634:
12605:
12543:
12533:
12523:
12449:
12379:
12350:
12327:
12323:
12300:
12296:
12253:
12176:
12109:
11697:
11693:
11639:
11612:
11552:
11515:
11478:
11247:
10966:
10946:
10924:
10904:
10778:
9920:
9152:
8653:
8568:
8449:
8227:
7912:. In addition to being graphed, the profile likelihood can also be used to compute
6100:
5777:
5466:
5460:
5172:
4651:
4587:
4563:
4543:
4523:
4438:
4418:
4332:
2931:
2247:
2227:
2196:
1432:
1294:
1254:
1173:
1153:
1133:
1113:
950:
923:
705:
645:
502:
482:
362:
105:
12971:
Unifying Political Methodology : the Likehood Theory of Statistical Inference
12135:
Kalbfleisch, J. D.; Sprott, D. A. (1973). "Marginal and Conditional Likelihoods".
2557:
15281:
15025:
14887:
14814:
14489:
14363:
14336:
14313:
14282:
13909:
13904:
13858:
13588:
13239:
13090:
12872:
12826:
12184:
12075:
GLIM 82: Proceedings of the International Conference on Generalised Linear Models
11914:
11738:
11519:
10869:
10785:
is then the natural logarithm of the likelihood function. Both terms are used in
8745:
8438:
8242:
8138:
6560:
4685:
2214:
941:
852:
347:
141:
14771:
12918:
12383:
6927:
Given a model, likelihood intervals can be compared to confidence intervals. If
15230:
15225:
13688:
13618:
13264:
12235:
10890:
10738:
9156:
8126:
6311:
6272:
4757:
370:
12609:
12354:
12180:
7916:
that often have better small-sample properties than those based on asymptotic
4331:
The above discussion of the likelihood for discrete random variables uses the
1074:{\displaystyle {\mathcal {L}}(\theta \mid x)=p_{\theta }(x)=P_{\theta }(X=x),}
682:
fixed, it is a probability density function, and when viewed as a function of
404:
The likelihood function, parameterized by a (possibly multivariate) parameter
15495:
15387:
15354:
15217:
15178:
14989:
14958:
14422:
14376:
13981:
13683:
13510:
13274:
13269:
12780:
A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, D. B. Rubin:
12499:
12477:
12070:
11483:
11464:
10893:, the likelihood when seen as a conditional density can be multiplied by the
10810:
10786:
10746:
9195:
9129:, determines the curvature of the likelihood surface, and thus indicates the
9030:
8582:
The equations defined by the stationary point of the score function serve as
8122:
7245:
6897:
of real values. If the region does comprise an interval, then it is called a
1497:
13126:
12639:
12454:
8459:
The log-likelihood function being plotted is used in the computation of the
1457:
15329:
15262:
15239:
15154:
14484:
13780:
13678:
13613:
13555:
13540:
13477:
13432:
12528:
11202:
8808:
8576:
7875:
6793:. This corresponds to standardizing the likelihood to have a maximum of 1.
6307:
5151:
4540:
is the index of the discrete probability mass corresponding to observation
255:
12803:
12230:
11847:
Hudson, D. J. (1971), "Interval estimation from the likelihood function",
8146:
is the logarithm of the likelihood function, often denoted by a lowercase
6904:
Likelihood intervals, and more generally likelihood regions, are used for
4338:
15372:
15334:
15017:
14918:
14780:
14593:
14560:
14052:
13969:
13964:
13608:
13565:
13545:
13525:
13515:
13284:
13006:
12989:
12431:
10806:
10802:
10711:
7959:
4646:
13016:
12148:
11153:
10829:
14218:
13698:
13398:
13329:
13279:
13254:
13174:
12463:
12370:
12167:
12121:
11871:
In All Likelihood: Statistical Modelling and Inference Using Likelihood
11564:
11543:
11492:
10889:, or vice versa. This is often the case in medical contexts. Following
8572:
8484:
8430:
8125:
in Bayesian statistics, but in likelihoodist statistics this is not an
6876:{\displaystyle \left\{\theta :R(\theta )\geq {\frac {p}{100}}\right\}.}
5167:
1559:
Consider a simple statistical model of a coin flip: a single parameter
12538:
7993:, is the product of the likelihoods of each of the individual events:
6125:
is the ratio of any two specified likelihoods, frequently written as:
5038:
vanishes, and if the likelihood function approaches a constant on the
2125:{\displaystyle {\mathcal {L}}(p_{\text{H}}=0.3\mid {\text{HH}})=0.09.}
1818:{\displaystyle {\mathcal {L}}(p_{\text{H}}=0.5\mid {\text{HH}})=0.25.}
14371:
14223:
13843:
13638:
13550:
13535:
13530:
13495:
12853:
12555:
11055:
10737:
The term "likelihood" has been in use in English since at least late
10703:{\textstyle \textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}
8567:. The basic way to maximize a differentiable function is to find the
8556:{\textstyle s_{n}(\theta )\equiv \nabla _{\theta }\ell _{n}(\theta )}
8259:
2519:
Relationship between the likelihood and probability density functions
1614:
13099:
Maximum Likelihood for Social Science : Strategies for Analysis
12399:
Maximum Likelihood for Social Science : Strategies for Analysis
12113:
11556:
10868:, although one can speak about the likelihood of any proposition or
9082:{\textstyle {\hat {\theta }}_{n}\xrightarrow {\text{p}} \theta _{0}}
9063:
13887:
13505:
13382:
13377:
13372:
9937:
looks rather daunting. Its logarithm is much simpler to work with:
8488:
4682:
1613:
can take on any value within the range 0.0 to 1.0. For a perfectly
338:
by calculating the probability of seeing that data under different
5121:{\displaystyle \lim _{\theta \to \partial \Theta }L(\theta )=0\;,}
4580:
amounts to maximizing the likelihood of the specific observation.
392:
of the parameter given the observed data, which is calculated via
15392:
15093:
12571:
Modeling with Data: Tools and Techniques for Scientific Computing
6950:
4280:
amounts to maximizing the likelihood of the specific observation
2061:{\displaystyle P({\text{HH}}\mid p_{\text{H}}=0.3)=0.3^{2}=0.09.}
1721:{\displaystyle P({\text{HH}}\mid p_{\text{H}}=0.5)=0.5^{2}=0.25.}
358:
10597:{\displaystyle {\widehat {\beta }}={\frac {\alpha }{\bar {x}}}.}
8433:
case). In the multivariate case, the concept generalizes into a
5150:
is unbounded. Mäkeläinen and co-authors prove this result using
15314:
14295:
14269:
14249:
13500:
13291:
13050:
Introductory Statistical Inference with the Likelihood Function
12590:
Mathematical Proceedings of the Cambridge Philosophical Society
11054:
In frequentist statistics, the likelihood function is itself a
10901:
density. More generally, the likelihood of an unknown quantity
1964:
Now suppose that the coin is not a fair coin, but instead that
11659:
Bayesian and Likelihood Methods in Statistics and Econometrics
10792:
5031:{\textstyle \;\nabla L\equiv \left_{i=1}^{n_{\mathrm {i} }}\;}
4560:, because maximizing the probability mass (or probability) at
1251:. The likelihood is the probability that a particular outcome
13143:
11463:
Mäkeläinen, Timo; Schmidt, Klaus; Styan, George P.H. (1981).
4513:{\displaystyle {\mathcal {L}}(\theta \mid x)=p_{k}(\theta ),}
4339:
Likelihoods for mixed continuous–discrete distributions
2343:{\displaystyle {\mathcal {L}}(\theta \mid x)=f_{\theta }(x),}
46:
13234:
12504:"On the mathematical foundations of theoretical statistics"
9483:{\textstyle {\boldsymbol {\eta }}({\boldsymbol {\theta }})}
6315:
6114:. For the statistical test to compare goodness of fit, see
6094:
5879:
This boundedness of the derivatives is needed to allow for
11991:. New York: Cambridge University Press. pp. 170â175.
11188:
paradigm, likelihood is interpreted within the context of
10797:
Among statisticians, there is no consensus about what the
8898:{\textstyle {\hat {\theta }}_{n}=s_{n}^{-1}(\mathbf {0} )}
7492:. Using this result, the maximum likelihood estimator for
12911:
Essential Statistical Inference : Theory and Methods
851:
is regarded as a fixed unknown quantity rather than as a
12847:
10897:
density of the parameter and then normalized, to give a
8112:
independent and identically distributed random variables
8110:
This is particularly important when the events are from
1730:
Equivalently, the likelihood of observing "HH" assuming
11962:. New York: Oxford University Press. pp. 267â269.
11582:. New York, NY: John Wiley & Sons. pp. 24â25.
12829:(1985). "Prediction and entropy". In Atkinson, A. C.;
11462:
10969:
10949:
10927:
10907:
10642:
10641:
10612:
10536:
10192:
10078:
9923:
9903:
9768:
9748:
9496:
9463:
9168:
9102:
9039:
8975:
8931:
8911:
8843:
8821:
8778:
8754:
8717:
8661:
8636:
8501:
8188:
7887:
7853:
7817:
7699:
7498:
7342:
7315:
7253:
7194:
7151:
7062:
7000:
6978:
6757:
6624:
6591:
6302:
The likelihood ratio is also of central importance in
6035:
5800:
5780:
5489:
5469:
5407:
5195:
5175:
5134:
5048:
4948:
4920:
4727:
4694:
4667:
4610:
4590:
4566:
4546:
4526:
4441:
4421:
4386:
4350:
4286:
4259:
3281:
2934:
2638:
2612:
2560:
2533:
2495:
2475:
2455:
2431:
2405:
2382:
2358:
2270:
2250:
2230:
2199:
2188:
2167:
2140:
1970:
1936:
1894:
1833:
1736:
1623:
1592:
1565:
1507:
1467:
1435:
1415:
1380:
1341:
1321:
1297:
1277:
1257:
1196:
1176:
1156:
1136:
1116:
1089:
973:
953:
926:
888:
868:
837:
802:
767:
732:
708:
688:
668:
648:
613:
505:
485:
410:
373:
at the maximum) gives an indication of the estimate's
15469:
11442:. New York: Cambridge University Press. p. 161.
11373:
An Introduction to Bayesian Inference in Econometrics
10558:
10240:
10100:
9945:
9790:
9691:
9652:
9623:
9531:
9341:
9206:
9022:{\textstyle s_{n}({\hat {\theta }}_{n})=\mathbf {0} }
8592:
8271:
8160:
7999:
7525:
6908:
within likelihoodist statistics: they are similar to
6828:
6661:
6418:
6392:
6361:
6330:
6131:
5893:
5517:
5225:
5076:
4765:
4462:
4116:
3686:
3543:
3318:
3310:
is the probability density function, it follows that
2954:
2715:
2292:
2078:
2005:
1771:
1665:
995:
569:
525:
439:
27:
Function related to statistics and probability theory
15056:
Autoregressive conditional heteroskedasticity (ARCH)
12672:
11958:(1993). "Concentrating the Loglikelihood Function".
11076:
is the count of parameters in some already-selected
7874:
that maximizes the likelihood function, creating an
7134:
can be determined explicitly, concentration reduces
1997:. Then the probability of two heads on two flips is
1271:
is observed when the true value of the parameter is
915:
12673:Bandyopadhyay, P. S.; Forster, M. R., eds. (2011).
11578:Greenberg, Edward; Webster, Charles E. Jr. (1983).
9151:of distributions, which include many of the common
9147:The log-likelihood is also particularly useful for
8704:{\textstyle s_{n}^{-1}:\mathbb {E} ^{d}\to \Theta }
6065:{\textstyle \,\left|\mathbf {I} (\theta )\right|\,}
2449:is not a probability density or mass function over
14518:
13068:
11922:
11850:Journal of the Royal Statistical Society, Series B
11805:Applied Statistical InferenceâLikelihood and Bayes
10975:
10955:
10933:
10913:
10702:
10627:
10596:
10542:
10520:
10224:
10186:If there are a number of independent observations
10176:
10084:
10068:To maximize the log-likelihood, we first take the
10058:
9929:
9909:
9887:
9774:
9754:
9733:
9708:
9669:
9631:
9605:
9511:
9482:
9447:
9323:
9186:
9117:
9081:
9021:
8961:
8917:
8897:
8829:
8799:
8760:
8732:
8703:
8644:
8622:
8555:
8441:. It has a relation to, but is distinct from, the
8403:
8198:
8166:
8099:
7939:. This form of conditioning is also the basis for
7900:
7866:
7832:
7799:
7685:
7511:
7484:
7328:
7301:
7236:
7176:
7126:
7048:
6986:
6875:
6785:
6743:
6639:
6606:
6538:
6412:odds, times the likelihood ratio. As an equation:
6398:
6374:
6343:
6244:
6064:
6017:
5871:
5786:
5766:
5503:
5475:
5451:
5393:
5212:
5181:
5142:
5120:
5062:
5030:
4934:
4902:
4748:
4713:
4673:
4616:
4596:
4572:
4552:
4532:
4512:
4447:
4427:
4407:
4372:
4299:
4272:
4245:
4100:
3670:
3523:
3302:
3265:
2940:
2920:
2701:
2624:
2598:
2546:
2507:
2481:
2461:
2441:
2417:
2388:
2364:
2342:
2276:
2256:
2236:
2205:
2177:
2153:
2124:
2060:
1989:
1953:
1922:
1888:given knowledge about the marginal probabilities
1876:
1817:
1755:
1720:
1642:
1605:
1578:
1547:
1485:
1441:
1421:
1401:
1366:
1327:
1303:
1283:
1263:
1202:
1182:
1162:
1142:
1122:
1095:
1073:
979:
959:
932:
900:
874:
843:
823:
788:
753:
714:
694:
674:
654:
634:
597:
555:
511:
491:
469:
416:
13071:Statistical Evidence : A Likelihood Paradigm
12848:Sakamoto, Y.; Ishiguro, M.; Kitagawa, G. (1986).
12794:Sox, H. C.; Higgins, M. C.; Owens, D. K. (2013),
12776:
12774:
12772:
12770:
12508:Philosophical Transactions of the Royal Society A
12134:
11982:
11953:
11656:
11580:Advanced Econometrics: A Bridge to the Literature
11432:
9313:
9250:
7092:
7070:
15493:
12097:
12049:. Princeton University Press. pp. 187â189.
11929:. Cambridge: Harvard University Press. pp.
11802:
11577:
11114:on a plot whose co-ordinates are the parameters
10717:
8962:{\textstyle \left\{{\hat {\theta }}_{n}\right\}}
7908:, the result of this procedure is also known as
5078:
3933:
3796:
3545:
3104:
556:{\displaystyle \theta \mapsto f(x\mid \theta ),}
342:values of the model. It is constructed from the
14604:Multivariate adaptive regression splines (MARS)
12866:
12793:
12714:
12712:
12710:
12708:
12316:Journal of the American Statistical Association
12313:
12288:Journal of the American Statistical Association
12208:. New York: John Wiley & Sons. p. 14.
12205:Geometrical Foundations of Asymptotic Inference
12018:. Norwich: W. H. Hutchins & Sons. pp.
10987:
6566:
1884:, a conclusion which could only be reached via
12973:. Cambridge University Press. pp. 59â94.
12767:
12756:
12754:
12752:
12750:
12722:(3rd ed., Oxford University Press 1983), §1.22
12588:Fisher, Ronald (1930). "Inverse Probability".
11983:Gourieroux, Christian; Monfort, Alain (1995).
11713:
11711:
11709:
11707:
11602:
11345:
9742:is an exponential family with two parameters,
7177:{\textstyle \mathbf {y} =\mathbf {X} \beta +u}
6951:Likelihoods that eliminate nuisance parameters
4641:, it suffices that the likelihood function is
2702:{\textstyle {\mathcal {L}}(\theta \mid x\in )}
2219:absolutely continuous probability distribution
598:{\displaystyle {\mathcal {L}}(\theta \mid x).}
13159:
12988:Richard, Mark; Vecer, Jan (1 February 2021).
12908:
12892:Statistical Inference Based on the Likelihood
12736:
12734:
12732:
12730:
12728:
11740:Statistical InferenceâBased on the likelihood
11295:
10820:
10635:denotes the maximum-likelihood estimate, and
6893:% likelihood region will usually comprise an
6786:{\textstyle {\mathcal {L}}({\hat {\theta }})}
4654:of the likelihood function plays a key role.
4253:and so maximizing the probability density at
1548:{\textstyle p_{\text{H}}^{2}(1-p_{\text{H}})}
303:
13089:
13033:. Oxford University Press. pp. 69â139.
12987:
12784:(3rd ed., Chapman & Hall/CRC 2014), §1.3
12705:
12397:Ward, Michael D.; Ahlquist, John S. (2018).
12396:
12340:
12248:
12246:
12228:
11819:
11817:
11815:
9580:
9555:
9401:
9365:
9291:
9255:
9181:
9169:
9089:. A similar result can be established using
8491:with respect to the parameter, known as the
5128:which may include the points at infinity if
1877:{\textstyle P(p_{\text{H}}=0.5\mid HH)=0.25}
12747:
12694:
12692:
12690:
12688:
12229:Papadopoulos, Alecos (September 25, 2013).
11717:
11704:
11505:
11395:
11346:Lehmann, Erich L.; Casella, George (1998).
11323:Frequentist and Bayesian Regression Methods
11144:
10793:Interpretations under different foundations
10745:in mathematical statistics was proposed by
9897:Finding the maximum likelihood estimate of
8925:. As a consequence there exists a sequence
8623:{\displaystyle s_{n}(\theta )=\mathbf {0} }
8563:, exists and allows for the application of
8206:for the likelihood. Because logarithms are
722:fixed, it is a likelihood function. In the
13204:
13166:
13152:
12725:
12492:
11887:
11296:Casella, George; Berger, Roger L. (2002).
10859:
7981:
6271:, the likelihood ratio is the basis for a
5865:
5114:
5056:
5049:
5027:
4949:
4899:
1657:, then the probability of observing HH is
1367:{\textstyle {\mathcal {L}}(\theta \mid x)}
1150:. Sometimes the probability of "the value
470:{\displaystyle x\mapsto f(x\mid \theta ),}
310:
296:
13817:
13015:
13005:
12638:
12537:
12527:
12453:
12243:
12137:SankhyÄ: The Indian Journal of Statistics
11890:"Generalized Linear Model - course notes"
11842:
11840:
11812:
11768:
11766:
11764:
11482:
11320:
11042:Learn how and when to remove this message
10055:
8720:
8685:
8463:(the gradient of the log-likelihood) and
7923:
6061:
6036:
5801:
5715:
5701:
5661:
5615:
5575:
5500:
5490:
5448:
5408:
5390:
5373:
5359:
5317:
5259:
5206:
5196:
5139:
5135:
4995:
4991:
4977:
4965:
4931:
4921:
4840:
4836:
4822:
4808:
4789:
4707:
4697:
4194:
4129:
4045:
4018:
3906:
3769:
3704:
3630:
3511:
3427:
3331:
3253:
3169:
3073:
2967:
2824:
2728:
1501:Figure 2. The likelihood function (
1461:Figure 1. The likelihood function (
1315:a probability density over the parameter
13095:"The Likelihood Function: A Deeper Dive"
12890:Azzalini, Adelchi (1996). "Likelihood".
12889:
12867:Burnham, K. P.; Anderson, D. R. (2002).
12744:(Cambridge University Press 2003), §4.1
12742:Probability Theory: The Logic of Science
12700:Probability and the Weighing of Evidence
12685:
12622:
12101:Journal of the Royal Statistical Society
11960:Estimation and Inference in Econometrics
11736:
11632:Journal of the Royal Statistical Society
11605:Journal of the Royal Statistical Society
11274:Exponential family § Interpretation
11005:This article includes a list of general
10741:. Its formal use to refer to a specific
9709:{\displaystyle A({\boldsymbol {\eta }})}
8129:because likelihoods are not integrated.
6095:Likelihood ratio and relative likelihood
4749:{\textstyle {\hat {\theta }}\in \Theta }
4627:
1496:
1456:
1409:, which is the posterior probability of
1291:, equivalent to the probability mass on
882:is the truth, given the observed sample
499:is a realization of the random variable
369:(often approximated by the likelihood's
183:Integrated nested Laplace approximations
13024:
12935:
12913:. New York: Springer. pp. 27â124.
12850:Akaike Information Criterion Statistics
12764:(Cambridge University Press 1980), §1.6
12568:
12252:
12201:
12073:(1982). "Direct Likelihood Inference".
12011:
11913:
11868:
11370:
11350:(2nd ed.). Springer. p. 444.
9699:
9625:
9593:
9559:
9539:
9473:
9465:
9414:
9377:
9369:
9349:
9304:
9267:
9259:
9220:
9136:
8478:
7966:estimation of the variance components.
7881:of the likelihood function for a given
6112:Likelihood ratios in diagnostic testing
5881:differentiation under the integral sign
14:
15494:
15130:KaplanâMeier estimator (product limit)
13066:
12825:
12653:
12587:
12498:
12476:
12202:Kass, Robert E.; Vos, Paul W. (1997).
12069:
12042:
12015:An Introduction to Likelihood Analysis
11846:
11837:
11803:Held, L.; SabanĂŠs BovĂŠ, D. S. (2014),
11761:
11540:
11325:(1st ed.). Springer. p. 36.
11300:(2nd ed.). Duxbury. p. 290.
11089:give an accurate approximation of the
8586:for the maximum likelihood estimator.
8114:, such as independent observations or
7946:
7791:
7746:
7642:
7567:
7436:
7391:
7138:of the original maximization problem.
5459:in order to ensure the existence of a
15203:
14770:
14517:
13816:
13586:
13203:
13147:
13047:
12798:(2nd ed.), Wiley, chapters 3â4,
12284:
11823:
11723:Probability and Statistical Inference
11411:
10722:
9632:{\displaystyle {\boldsymbol {\eta }}}
7969:
7920:calculated from the full likelihood.
6966:
5883:. And lastly, it is assumed that the
424:, is usually defined differently for
365:for the unknown parameter, while the
15440:
15140:Accelerated failure time (AFT) model
12961:
12894:. Chapman and Hall. pp. 17â50.
12666:
12430:
11681:
11629:
11417:
11148:
10991:
10824:
9153:parametric probability distributions
8425:of the log-likelihood is called the
6796:
6614:. Relative plausibilities of other
6559:to assess the value of performing a
5452:{\textstyle \,r,s,t=1,2,\ldots ,k\,}
5213:{\textstyle \,\theta \in \Theta \,,}
4317:measure-theoretic probability theory
1827:This is not the same as saying that
361:the likelihood function serves as a
15452:
14735:Analysis of variance (ANOVA, anova)
13587:
12967:"The Likelihood Model of Inference"
12367:
12161:
11375:. New York: Wiley. pp. 13â14.
9096:The second derivative evaluated at
8837:with probability going to one, and
7928:Sometimes it is possible to find a
6254:The likelihood ratio is central to
6105:
6091:of the posterior in large samples.
6080:properties may need to be assumed.
5504:{\textstyle \,\theta \in \Theta \,}
4935:{\textstyle \,\theta \in \Theta \,}
2189:Continuous probability distribution
24:
14830:CochranâMantelâHaenszel statistics
13456:Pearson product-moment correlation
12883:
11985:"Concentrated Likelihood Function"
11919:"Concentrated Likelihood Function"
11644:10.1111/j.2517-6161.1985.tb01384.x
11617:10.1111/j.2517-6161.1979.tb01071.x
11011:it lacks sufficient corresponding
10450:
10415:
10404:
10383:
10348:
10337:
10314:
10260:
10249:
10143:
10115:
10104:
10004:
9954:
9835:
9793:
9679:, minus the normalization factor (
8768:is the parameter space. Using the
8755:
8698:
8525:
8483:If the log-likelihood function is
8354:
8329:
8301:
8283:
8214:, in particular since most common
8191:
8069:
8041:
8000:
7986:The likelihood, given two or more
7184:, the coefficient vector could be
7145:with normally distributed errors,
6760:
6706:
6682:
6495:
6209:
6178:
6132:
6008:
5982:
5968:
5946:
5932:
5924:
5919:
5862:
5846:
5815:
5810:
5716:
5702:
5688:
5674:
5616:
5602:
5588:
5533:
5525:
5497:
5374:
5360:
5346:
5326:
5301:
5288:
5268:
5243:
5229:
5203:
5136:
5091:
5088:
5063:{\textstyle \;\partial \Theta \;,}
5053:
5050:
5019:
4978:
4969:
4950:
4928:
4891:
4876:
4823:
4809:
4794:
4743:
4668:
4465:
4201:
4198:
4195:
4191:
4188:
4185:
4152:
4136:
4133:
4130:
4126:
4123:
4120:
4052:
4049:
4046:
4042:
4039:
4036:
3913:
3910:
3907:
3903:
3900:
3897:
3820:
3776:
3773:
3770:
3766:
3763:
3760:
3727:
3711:
3708:
3705:
3701:
3698:
3695:
3434:
3431:
3428:
3424:
3421:
3418:
3354:
3338:
3335:
3332:
3328:
3325:
3322:
3176:
3173:
3170:
3166:
3163:
3160:
3080:
3077:
3074:
3070:
3067:
3064:
3000:
2974:
2971:
2968:
2964:
2961:
2958:
2948:is positive and constant. Because
2857:
2831:
2828:
2825:
2821:
2818:
2815:
2751:
2735:
2732:
2729:
2725:
2722:
2719:
2641:
2554:, the likelihood for the interval
2434:
2295:
2170:
2134:More generally, for each value of
2081:
1774:
1344:
998:
572:
384:, the estimate of interest is the
25:
15518:
13127:Likelihood function at Planetmath
13120:
12418:Shorter Oxford English Dictionary
12104:. Series C (Applied Statistics).
11989:Statistics and Econometric Models
11439:Statistics and Econometric Models
11422:(2nd ed.). Springer. §4.4.1.
8176:, to contrast with the uppercase
8132:
7336:yields an optimal value function
6072:is finite. This ensures that the
916:Discrete probability distribution
388:of the likelihood, the so-called
15479:
15451:
15439:
15427:
15414:
15413:
15204:
13031:Parametric Statistical Inference
11774:Statistical Inference in Science
11152:
11097:probability of having happened.
10996:
10828:
10628:{\textstyle {\widehat {\beta }}}
10225:{\textstyle x_{1},\ldots ,x_{n}}
9654:
9567:
9388:
9278:
9187:{\textstyle \langle -,-\rangle }
9015:
8888:
8823:
8638:
8616:
7820:
7780:
7754:
7735:
7717:
7702:
7679:
7663:
7654:
7631:
7605:
7588:
7579:
7556:
7457:
7448:
7425:
7399:
7380:
7284:
7269:
7255:
7161:
7153:
6043:
5895:
4767:
4714:{\textstyle \mathbb {R} ^{k}\,,}
1923:{\textstyle P(p_{\text{H}}=0.5)}
277:
193:Approximate Bayesian computation
45:
15089:Least-squares spectral analysis
12860:
12841:
12819:
12787:
12647:
12616:
12581:
12562:
12470:
12424:
12409:
12390:
12361:
12334:
12307:
12278:
12222:
12195:
12155:
12128:
12091:
12063:
12046:Ecological Models and Data in R
12036:
12005:
11976:
11947:
11907:
11881:
11862:
11796:
11779:
11730:
11675:
11650:
11623:
11596:
11571:
11534:
11266:
10921:given another unknown quantity
9734:Example: the gamma distribution
9724:and the log-partition function
9670:{\displaystyle \mathbf {T} (x)}
7302:{\textstyle \mathbf {X} =\left}
6987:{\textstyle \mathbf {\theta } }
5665:
5579:
5321:
5263:
3535:fundamental theorem of calculus
2264:) which depends on a parameter
219:Maximum a posteriori estimation
14070:Mean-unbiased minimum-variance
13173:
13075:. London: Chapman & Hall.
12945:Johns Hopkins University Press
12328:10.1080/01621459.1975.10480321
12301:10.1080/01621459.1977.10479926
12262:Johns Hopkins University Press
12165:(1975). "Partial likelihood".
11698:10.1080/00031305.1982.10482817
11661:. Elsevier. pp. 473â488.
11499:
11456:
11426:
11389:
11364:
11339:
11314:
11289:
10649:
10584:
10445:
10420:
10378:
10353:
10309:
10265:
10138:
10120:
10031:
10019:
10013:
10007:
9977:
9959:
9844:
9838:
9816:
9798:
9703:
9695:
9664:
9658:
9597:
9589:
9577:
9571:
9549:
9535:
9506:
9500:
9477:
9469:
9439:
9433:
9418:
9410:
9398:
9392:
9381:
9373:
9359:
9345:
9308:
9300:
9288:
9282:
9271:
9263:
9239:
9233:
9224:
9210:
9109:
9047:
9008:
8996:
8986:
8943:
8892:
8884:
8851:
8695:
8609:
8603:
8550:
8544:
8518:
8512:
8454:statistical hypothesis testing
8395:
8389:
8380:
8374:
8365:
8359:
8340:
8334:
8312:
8306:
8294:
8288:
8091:
8072:
8063:
8044:
8035:
8003:
7840:. This result is known as the
7533:
7366:
7353:
7309:). Maximizing with respect to
6912:in frequentist statistics and
6889:is a single real parameter, a
6849:
6843:
6780:
6774:
6765:
6732:
6720:
6711:
6699:
6687:
6671:
6665:
6631:
6598:
6557:are used in diagnostic testing
6530:
6498:
6489:
6463:
6454:
6422:
6318:, Bayes' rule states that the
6233:
6214:
6202:
6183:
6167:
6135:
6053:
6047:
5905:
5899:
5842:
5836:
5761:
5755:
5658:
5652:
5572:
5566:
5105:
5099:
5085:
4777:
4771:
4734:
4721:there exists a unique maximum
4504:
4498:
4482:
4470:
4402:
4390:
4367:
4361:
4237:
4218:
4176:
4157:
4088:
4069:
4015:
4003:
3940:
3875:
3872:
3840:
3825:
3803:
3751:
3732:
3662:
3643:
3627:
3615:
3552:
3508:
3496:
3409:
3406:
3374:
3359:
3297:
3285:
3250:
3238:
3151:
3107:
3055:
3052:
3020:
3005:
2912:
2909:
2877:
2862:
2806:
2803:
2771:
2756:
2696:
2693:
2661:
2646:
2593:
2561:
2469:, despite being a function of
2334:
2328:
2312:
2300:
2113:
2086:
2036:
2009:
1948:
1940:
1917:
1898:
1865:
1837:
1806:
1779:
1696:
1669:
1542:
1523:
1396:
1384:
1374:, should not be confused with
1361:
1349:
1065:
1053:
1037:
1031:
1015:
1003:
818:
806:
783:
771:
761:is often avoided and instead
748:
736:
629:
617:
589:
577:
547:
535:
529:
461:
449:
443:
344:joint probability distribution
13:
1:
15383:Geographic information system
14599:Simultaneous equations models
11634:. Series B (Methodological).
11607:. Series B (Methodological).
11282:
10718:Background and interpretation
9782:. The likelihood function is
9118:{\textstyle {\hat {\theta }}}
8733:{\textstyle \mathbb {E} ^{d}}
8469:maximum likelihood estimation
8248:maximum likelihood estimation
8212:maximum likelihood estimation
7833:{\textstyle \mathbf {X} _{2}}
7056:, and where a correspondence
6994:that can be partitioned into
6640:{\textstyle {\hat {\theta }}}
6607:{\textstyle {\hat {\theta }}}
6087:, and therefore to justify a
4661:-dimensional parameter space
4604:, but not with the parameter
4408:{\textstyle f(x\mid \theta )}
4310:
3303:{\textstyle f(x\mid \theta )}
1990:{\textstyle p_{\text{H}}=0.3}
1756:{\textstyle p_{\text{H}}=0.5}
1643:{\textstyle p_{\text{H}}=0.5}
1486:{\textstyle p_{\text{H}}^{2}}
1402:{\textstyle P(\theta \mid x)}
862:specify the probability that
858:The likelihood function does
754:{\textstyle f(x\mid \theta )}
635:{\textstyle f(x\mid \theta )}
519:, the likelihood function is
399:
355:maximum likelihood estimation
14566:Coefficient of determination
14177:Uniformly most powerful test
13093:; Ahlquist, John S. (2018).
12343:Communications in Statistics
12077:. Springer. pp. 76â86.
12043:Bolker, Benjamin M. (2008).
11520:10.1080/02331934.2010.527973
11243:Principle of maximum entropy
10988:Likelihoodist interpretation
10751:method of maximum likelihood
9917:for a single observed value
8905:is a consistent estimate of
6805:is the set of all values of
6567:Relative likelihood function
4435:'s added to the integral of
2352:considered as a function of
1083:considered as a function of
126:Principle of maximum entropy
7:
15135:Proportional hazards models
15079:Spectral density estimation
15061:Vector autoregression (VAR)
14495:Maximum posterior estimator
13727:Randomized controlled trial
12919:10.1007/978-1-4614-4818-1_2
12835:A Celebration of Statistics
11195:
8199:{\textstyle {\mathcal {L}}}
7964:residual maximum likelihood
7937:hypergeometric distribution
7842:FrischâWaughâLovell theorem
6579:maximum likelihood estimate
6306:, where it is known as the
4415:, where the sum of all the
4373:{\textstyle p_{k}(\theta )}
2632:is a constant, is given by
2442:{\textstyle {\mathcal {L}}}
2178:{\textstyle {\mathcal {L}}}
1954:{\textstyle P({\text{HH}})}
642:is viewed as a function of
96:Bernsteinâvon Mises theorem
10:
15523:
14895:Multivariate distributions
13315:Average absolute deviation
13103:Cambridge University Press
13048:Rohde, Charles A. (2014).
12837:. Springer. pp. 1â24.
12575:Princeton University Press
12403:Cambridge University Press
11791:Cambridge University Press
11776:, Springer (chap. 2).
11348:Theory of Point Estimation
10821:Frequentist interpretation
10726:
9140:
8136:
7976:proportional hazards model
7950:
6570:
6322:odds of two alternatives,
6109:
6098:
1452:
831:are used to indicate that
15409:
15363:
15300:
15253:
15216:
15212:
15199:
15171:
15153:
15120:
15111:
15069:
15016:
14977:
14926:
14917:
14883:Structural equation model
14838:
14795:
14791:
14766:
14725:
14691:
14645:
14612:
14574:
14541:
14537:
14513:
14453:
14362:
14281:
14245:
14236:
14219:Score/Lagrange multiplier
14204:
14157:
14102:
14028:
14019:
13829:
13825:
13812:
13771:
13745:
13697:
13652:
13634:Sample size determination
13599:
13595:
13582:
13486:
13441:
13415:
13397:
13353:
13305:
13225:
13216:
13212:
13199:
13181:
12610:10.1017/S0305004100016297
12384:10.1093/biomet/47.1-2.203
12355:10.1080/03610928208828325
11685:The American Statistician
11436:; Monfort, Alain (1995).
8830:{\textstyle \mathbf {0} }
8645:{\textstyle \mathbf {0} }
8443:support of a distribution
8216:probability distributions
8116:sampling with replacement
7237:{\textstyle \beta =\left}
5463:. Second, for almost all
4758:matrix of second partials
2154:{\textstyle p_{\text{H}}}
1606:{\textstyle p_{\text{H}}}
1579:{\textstyle p_{\text{H}}}
967:depending on a parameter
946:probability mass function
824:{\textstyle f(x,\theta )}
789:{\textstyle f(x;\theta )}
429:probability distributions
326:(often simply called the
121:Principle of indifference
15378:Environmental statistics
14900:Elliptical distributions
14693:Generalized linear model
14622:Simple linear regression
14392:HodgesâLehmann estimator
13849:Probability distribution
13758:Stochastic approximation
13320:Coefficient of variation
13067:Royall, Richard (1997).
12679:North-Holland Publishing
12675:Philosophy of Statistics
12012:Pickles, Andrew (1985).
11371:Zellner, Arnold (1971).
11259:
11233:Likelihoodist statistics
11145:AIC-based interpretation
10799:foundation of statistics
8770:inverse function theorem
8416:
8234:plays a key role in the
6941:chi-squared distribution
6256:likelihoodist statistics
5143:{\textstyle \,\Theta \,}
5042:of the parameter space,
4325:RadonâNikodym derivative
1190:for the parameter value
855:being conditioned on.
173:Markov chain Monte Carlo
15038:Cross-correlation (XCF)
14646:Non-standard predictors
14080:LehmannâScheffĂŠ theorem
13753:Adaptive clinical trial
13025:Lindsey, J. K. (1996).
12796:Medical Decision Making
12181:10.1093/biomet/62.2.269
11875:Oxford University Press
11826:Mathematical Statistics
11785:Davison, A. C. (2008),
11420:Mathematical Statistics
11401:Probability and Measure
11321:Wakefield, Jon (2013).
11213:Conditional probability
11026:more precise citations.
10941:is proportional to the
10860:Bayesian interpretation
8800:{\textstyle s_{n}^{-1}}
8772:, it can be shown that
8448:The term was coined by
8224:logarithmically concave
8144:Log-likelihood function
7982:Products of likelihoods
7901:{\textstyle \beta _{1}}
7867:{\textstyle \beta _{2}}
7519:can then be derived as
7512:{\textstyle \beta _{1}}
7329:{\textstyle \beta _{2}}
6553:evidence-based medicine
6287:test for comparing two
6076:has a finite variance.
1130:of the random variable
426:discrete and continuous
359:argument that maximizes
178:Laplace's approximation
165:Posterior approximation
15434:Mathematics portal
15255:Engineering statistics
15163:NelsonâAalen estimator
14740:Analysis of covariance
14627:Ordinary least squares
14551:Pearson product-moment
13955:Statistical functional
13866:Empirical distribution
13699:Controlled experiments
13428:Frequency distribution
13206:Descriptive statistics
12782:Bayesian Data Analysis
12529:10.1098/rsta.1922.0009
12345:. Theory and Methods.
11869:Pawitan, Yudi (2001).
11772:Sprott, D. A. (2000),
11484:10.1214/aos/1176345516
11091:frequency distribution
10977:
10957:
10935:
10915:
10837:This section is empty.
10769:
10760:
10733:History of probability
10704:
10688:
10629:
10598:
10544:
10522:
10500:
10226:
10178:
10086:
10060:
9931:
9911:
9889:
9776:
9756:
9710:
9681:log-partition function
9671:
9633:
9607:
9513:
9484:
9449:
9325:
9188:
9119:
9083:
9023:
8963:
8919:
8899:
8831:
8801:
8762:
8734:
8705:
8646:
8624:
8571:(the points where the
8557:
8473:likelihood-ratio tests
8405:
8200:
8168:
8101:
7924:Conditional likelihood
7902:
7868:
7834:
7801:
7687:
7513:
7486:
7330:
7303:
7244:(and consequently the
7238:
7178:
7128:
7050:
6988:
6877:
6787:
6745:
6641:
6608:
6540:
6400:
6376:
6345:
6246:
6066:
6019:
5873:
5788:
5768:
5505:
5477:
5453:
5395:
5214:
5183:
5144:
5122:
5064:
5032:
4942:at which the gradient
4936:
4904:
4750:
4715:
4675:
4618:
4598:
4574:
4554:
4534:
4514:
4449:
4429:
4409:
4374:
4301:
4274:
4247:
4102:
3672:
3525:
3304:
3267:
2942:
2922:
2703:
2626:
2600:
2548:
2509:
2489:given the observation
2483:
2463:
2443:
2419:
2390:
2366:
2344:
2278:
2258:
2238:
2207:
2179:
2155:
2126:
2062:
1991:
1955:
1924:
1878:
1819:
1757:
1722:
1644:
1607:
1580:
1556:
1549:
1494:
1487:
1443:
1423:
1403:
1368:
1329:
1305:
1285:
1265:
1204:
1184:
1164:
1144:
1124:
1097:
1075:
981:
961:
934:
902:
876:
845:
825:
790:
755:
716:
696:
676:
656:
636:
599:
557:
513:
493:
471:
418:
330:) measures how well a
284:Mathematics portal
227:Evidence approximation
15350:Population statistics
15292:System identification
15026:Autocorrelation (ACF)
14954:Exponential smoothing
14868:Discriminant analysis
14863:Canonical correlation
14727:Partition of variance
14589:Regression validation
14433:(JonckheereâTerpstra)
14332:Likelihood-ratio test
14021:Frequentist inference
13933:Locationâscale family
13854:Sampling distribution
13819:Statistical inference
13786:Cross-sectional study
13773:Observational studies
13732:Randomized experiment
13561:Stem-and-leaf display
13363:Central limit theorem
12943:(Expanded ed.).
12804:10.1002/9781118341544
12720:Theory of Probability
12640:10.1214/ss/1030037905
12569:Klemens, Ben (2008).
12455:10.1214/ss/1009212248
11925:Advanced Econometrics
11824:Rossi, R. J. (2018),
11737:Azzalini, A. (1996),
11434:GouriĂŠroux, Christian
11405:John Wiley & Sons
11298:Statistical Inference
11228:Likelihood-ratio test
10978:
10958:
10936:
10916:
10899:posterior probability
10764:
10755:
10729:History of statistics
10714:of the observations.
10705:
10668:
10630:
10599:
10545:
10523:
10480:
10227:
10179:
10087:
10061:
9932:
9912:
9890:
9777:
9757:
9711:
9672:
9634:
9608:
9521:change of coordinates
9519:each correspond to a
9514:
9485:
9450:
9326:
9189:
9141:Further information:
9120:
9084:
9024:
8964:
8920:
8900:
8832:
8802:
8763:
8735:
8706:
8647:
8625:
8565:differential calculus
8558:
8406:
8201:
8169:
8167:{\displaystyle \ell }
8102:
7903:
7869:
7835:
7802:
7688:
7514:
7487:
7331:
7304:
7239:
7179:
7129:
7051:
6989:
6922:posterior probability
6878:
6788:
6746:
6642:
6609:
6541:
6401:
6377:
6375:{\displaystyle A_{2}}
6346:
6344:{\displaystyle A_{1}}
6314:. Stated in terms of
6277:likelihood-ratio test
6269:frequentist inference
6247:
6116:Likelihood-ratio test
6089:Laplace approximation
6085:posterior probability
6067:
6020:
5874:
5789:
5769:
5506:
5478:
5454:
5396:
5215:
5184:
5156:mountain pass theorem
5145:
5123:
5065:
5033:
4937:
4905:
4751:
4716:
4676:
4639:extreme value theorem
4628:Regularity conditions
4619:
4599:
4575:
4555:
4535:
4515:
4450:
4430:
4410:
4375:
4302:
4275:
4248:
4103:
3673:
3526:
3305:
3268:
2943:
2923:
2704:
2627:
2601:
2549:
2510:
2484:
2464:
2444:
2420:
2391:
2367:
2345:
2279:
2259:
2239:
2208:
2180:
2156:
2127:
2063:
1992:
1956:
1925:
1879:
1820:
1758:
1723:
1645:
1608:
1581:
1550:
1500:
1488:
1460:
1444:
1424:
1404:
1369:
1330:
1306:
1286:
1266:
1205:
1185:
1165:
1145:
1125:
1098:
1076:
982:
962:
935:
903:
877:
846:
826:
791:
756:
717:
697:
677:
657:
637:
607:In other words, when
600:
558:
514:
494:
472:
419:
390:posterior probability
188:Variational inference
15273:Probabilistic design
14858:Principal components
14701:Exponential families
14653:Nonlinear regression
14632:General linear model
14594:Mixed effects models
14584:Errors and residuals
14561:Confounding variable
14463:Bayesian probability
14441:Van der Waerden test
14431:Ordered alternative
14196:Multiple comparisons
14075:RaoâBlackwellization
14038:Estimating equations
13994:Statistical distance
13712:Factorial experiment
13245:Arithmetic-Geometric
13052:. Berlin: Springer.
13007:10.3390/risks9020031
12702:(Griffin 1950), §6.1
12656:Statistical Evidence
12514:(594â604): 309â368.
11896:. pp. Chapter 5
11892:. Taichung, Taiwan:
11470:Annals of Statistics
11397:Billingsley, Patrick
11223:Likelihood principle
11218:Empirical likelihood
10967:
10947:
10925:
10905:
10639:
10610:
10556:
10534:
10238:
10190:
10098:
10076:
9943:
9921:
9901:
9788:
9766:
9755:{\textstyle \alpha }
9746:
9689:
9650:
9643:sufficient statistic
9621:
9529:
9494:
9461:
9339:
9204:
9166:
9149:exponential families
9137:Exponential families
9100:
9037:
8973:
8929:
8918:{\textstyle \theta }
8909:
8841:
8819:
8776:
8761:{\textstyle \Theta }
8752:
8715:
8659:
8634:
8590:
8584:estimating equations
8499:
8479:Likelihood equations
8269:
8186:
8158:
7997:
7930:sufficient statistic
7914:confidence intervals
7885:
7851:
7815:
7697:
7523:
7496:
7340:
7313:
7251:
7192:
7149:
7136:computational burden
7060:
6998:
6976:
6918:coverage probability
6910:confidence intervals
6826:
6755:
6659:
6622:
6589:
6555:, likelihood ratios
6416:
6390:
6359:
6328:
6281:NeymanâPearson lemma
6129:
6033:
5891:
5798:
5778:
5515:
5487:
5467:
5405:
5223:
5193:
5173:
5132:
5074:
5046:
4946:
4918:
4763:
4725:
4692:
4674:{\textstyle \Theta }
4665:
4617:{\textstyle \theta }
4608:
4588:
4564:
4544:
4524:
4460:
4439:
4419:
4384:
4348:
4284:
4257:
4114:
3684:
3541:
3316:
3279:
2952:
2932:
2713:
2636:
2610:
2558:
2531:
2493:
2482:{\textstyle \theta }
2473:
2462:{\textstyle \theta }
2453:
2429:
2403:
2389:{\textstyle \theta }
2380:
2365:{\textstyle \theta }
2356:
2290:
2284:. Then the function
2277:{\textstyle \theta }
2268:
2248:
2228:
2197:
2165:
2138:
2076:
2003:
1968:
1934:
1892:
1831:
1769:
1734:
1663:
1621:
1590:
1563:
1505:
1465:
1433:
1422:{\textstyle \theta }
1413:
1378:
1339:
1328:{\textstyle \theta }
1319:
1295:
1284:{\textstyle \theta }
1275:
1255:
1203:{\textstyle \theta }
1194:
1174:
1154:
1134:
1114:
1096:{\textstyle \theta }
1087:
993:
987:. Then the function
980:{\textstyle \theta }
971:
951:
924:
910:prosecutor's fallacy
886:
875:{\textstyle \theta }
866:
844:{\textstyle \theta }
835:
800:
765:
730:
724:frequentist paradigm
706:
695:{\textstyle \theta }
686:
675:{\textstyle \theta }
666:
646:
611:
567:
523:
503:
483:
437:
417:{\textstyle \theta }
408:
266:Posterior predictive
235:Evidence lower bound
116:Likelihood principle
86:Bayesian probability
15507:Bayesian statistics
15345:Official statistics
15268:Methods engineering
14949:Seasonal adjustment
14717:Poisson regressions
14637:Bayesian regression
14576:Regression analysis
14556:Partial correlation
14528:Regression analysis
14127:Prediction interval
14122:Likelihood interval
14112:Confidence interval
14104:Interval estimation
14065:Unbiased estimators
13883:Model specification
13763:Up-and-down designs
13451:Partial correlation
13407:Index of dispersion
13325:Interquartile range
12654:Royall, R. (1997).
12626:Statistical Science
12602:1930PCPS...26..528F
12520:1922RSPTA.222..309F
12441:Statistical Science
11956:MacKinnon, James G.
11954:Davidson, Russell;
11407:. pp. 422â423.
11208:Conditional entropy
10882:conditional density
10878:marginal likelihood
10773:inverse probability
10753:". Quoting Fisher:
10543:{\textstyle \beta }
10085:{\textstyle \beta }
9910:{\textstyle \beta }
9775:{\textstyle \beta }
9067:
8883:
8796:
8679:
8256:information content
8208:strictly increasing
7953:Marginal likelihood
7947:Marginal likelihood
7941:Fisher's exact test
7796:
7751:
7647:
7572:
7441:
7396:
7141:For instance, in a
6957:nuisance parameters
6906:interval estimation
6899:likelihood interval
6814:% likelihood region
6649:relative likelihood
6573:Relative likelihood
6283:, this is the most
5928:
5819:
5026:
4898:
3999:
3611:
3492:
3234:
2625:{\textstyle h>0}
2525:probability density
2374:likelihood function
1522:
1482:
1105:likelihood function
382:Bayesian statistics
324:likelihood function
39:Bayesian statistics
33:Part of a series on
15365:Spatial statistics
15245:Medical statistics
15145:First hitting time
15099:Whittle likelihood
14750:Degrees of freedom
14745:Multivariate ANOVA
14678:Heteroscedasticity
14490:Bayesian estimator
14455:Bayesian inference
14304:KolmogorovâSmirnov
14189:Randomization test
14159:Testing hypotheses
14132:Tolerance interval
14043:Maximum likelihood
13938:Exponential family
13871:Density estimation
13831:Statistical theory
13791:Natural experiment
13737:Scientific control
13654:Survey methodology
13340:Standard deviation
13105:. pp. 21â28.
12660:Chapman & Hall
11894:Tunghai University
11787:Statistical Models
11745:Chapman & Hall
11719:Kalbfleisch, J. G.
11514:(8â9): 1121â1159.
11418:Shao, Jun (2003).
11403:(Third ed.).
11253:Score (statistics)
11238:Maximum likelihood
11190:information theory
11164:. You can help by
10973:
10953:
10931:
10911:
10866:Bayesian inference
10723:Historical remarks
10700:
10699:
10625:
10594:
10540:
10518:
10516:
10222:
10174:
10082:
10070:partial derivative
10056:
9927:
9907:
9885:
9772:
9752:
9740:gamma distribution
9706:
9667:
9629:
9603:
9509:
9480:
9445:
9321:
9184:
9143:Exponential family
9127:Fisher information
9115:
9079:
9019:
8959:
8915:
8895:
8866:
8827:
8797:
8779:
8758:
8730:
8701:
8662:
8642:
8620:
8553:
8465:Fisher information
8452:in the context of
8401:
8232:objective function
8220:exponential family
8196:
8164:
8097:
7970:Partial likelihood
7910:profile likelihood
7898:
7864:
7830:
7797:
7778:
7733:
7683:
7629:
7554:
7509:
7482:
7423:
7378:
7326:
7299:
7234:
7174:
7124:
7046:
6984:
6967:Profile likelihood
6914:credible intervals
6873:
6783:
6741:
6655:is defined to be
6637:
6604:
6581:for the parameter
6536:
6396:
6372:
6341:
6304:Bayesian inference
6293:significance level
6242:
6062:
6015:
5911:
5885:information matrix
5869:
5802:
5784:
5764:
5501:
5473:
5449:
5391:
5210:
5179:
5140:
5118:
5095:
5060:
5028:
4959:
4932:
4900:
4783:
4746:
4711:
4671:
4614:
4594:
4570:
4550:
4530:
4510:
4445:
4425:
4405:
4370:
4323:is defined as the
4300:{\textstyle x_{j}}
4297:
4273:{\textstyle x_{j}}
4270:
4243:
4211:
4146:
4098:
4096:
4062:
3965:
3954:
3923:
3817:
3786:
3721:
3668:
3577:
3566:
3521:
3458:
3444:
3348:
3300:
3263:
3200:
3186:
3090:
2984:
2938:
2918:
2841:
2745:
2699:
2622:
2596:
2547:{\textstyle x_{j}}
2544:
2505:
2479:
2459:
2439:
2415:
2386:
2362:
2340:
2274:
2254:
2234:
2203:
2175:
2151:
2122:
2058:
1987:
1951:
1920:
1874:
1815:
1753:
1718:
1640:
1603:
1576:
1557:
1545:
1508:
1495:
1483:
1468:
1439:
1419:
1399:
1364:
1335:. The likelihood,
1325:
1301:
1281:
1261:
1200:
1180:
1160:
1140:
1120:
1093:
1071:
977:
957:
930:
898:
872:
841:
821:
786:
751:
712:
692:
672:
652:
632:
595:
553:
509:
489:
467:
414:
367:Fisher information
209:Bayesian estimator
157:Hierarchical model
81:Bayesian inference
15467:
15466:
15405:
15404:
15401:
15400:
15340:National accounts
15310:Actuarial science
15302:Social statistics
15195:
15194:
15191:
15190:
15187:
15186:
15122:Survival function
15107:
15106:
14969:Granger causality
14810:Contingency table
14785:Survival analysis
14762:
14761:
14758:
14757:
14614:Linear regression
14509:
14508:
14505:
14504:
14480:Credible interval
14449:
14448:
14232:
14231:
14048:Method of moments
13917:Parametric family
13878:Statistical model
13808:
13807:
13804:
13803:
13722:Random assignment
13644:Statistical power
13578:
13577:
13574:
13573:
13423:Contingency table
13393:
13392:
13260:Generalized/power
13112:978-1-316-63682-4
13059:978-3-319-10460-7
12937:Edwards, A. W. F.
12928:978-1-4614-4817-4
12405:. pp. 25â27.
12349:(13): 1505â1510.
12254:Edwards, A. W. F.
12056:978-0-691-12522-0
11998:978-0-521-40551-5
11969:978-0-19-506011-9
11940:978-0-674-00560-0
11332:978-1-4419-0925-1
11182:
11181:
11112:confidence region
11078:statistical model
11052:
11051:
11044:
10895:prior probability
10874:statistical model
10857:
10856:
10666:
10652:
10622:
10589:
10587:
10568:
10475:
10457:
10390:
10321:
10163:
10150:
9848:
9512:{\textstyle h(x)}
9133:of the estimate.
9112:
9068:
9066:
9050:
8999:
8946:
8854:
8813:open neighborhood
8569:stationary points
8316:
7809:projection matrix
7536:
7143:linear regression
7095:
7073:
6920:(frequentism) or
6863:
6820:is defined to be
6803:likelihood region
6797:Likelihood region
6777:
6736:
6723:
6634:
6601:
6399:{\displaystyle B}
6384:, given an event
6310:, and is used in
6289:simple hypotheses
6261:law of likelihood
6237:
6027:positive definite
6006:
6000:
5996:
5964:
5960:
5730:
5630:
5547:
5388:
5315:
5257:
5161:In the proofs of
5077:
4993:
4912:negative definite
4838:
4737:
4681:assumed to be an
4182:
4117:
4033:
3963:
3932:
3894:
3795:
3757:
3692:
3575:
3544:
3456:
3415:
3319:
3198:
3157:
3102:
3061:
2996:
2955:
2853:
2812:
2716:
2148:
2111:
2096:
2027:
2015:
1978:
1946:
1908:
1847:
1804:
1789:
1744:
1687:
1675:
1631:
1600:
1573:
1539:
1515:
1475:
332:statistical model
320:
319:
214:Credible interval
147:Linear regression
16:(Redirected from
15514:
15484:
15483:
15475:
15455:
15454:
15443:
15442:
15432:
15431:
15417:
15416:
15320:Crime statistics
15214:
15213:
15201:
15200:
15118:
15117:
15084:Fourier analysis
15071:Frequency domain
15051:
14998:
14964:Structural break
14924:
14923:
14873:Cluster analysis
14820:Log-linear model
14793:
14792:
14768:
14767:
14709:
14683:Homoscedasticity
14539:
14538:
14515:
14514:
14434:
14426:
14418:
14417:(KruskalâWallis)
14402:
14387:
14342:Cross validation
14327:
14309:AndersonâDarling
14256:
14243:
14242:
14214:Likelihood-ratio
14206:Parametric tests
14184:Permutation test
14167:1- & 2-tails
14058:Minimum distance
14030:Point estimation
14026:
14025:
13977:Optimal decision
13928:
13827:
13826:
13814:
13813:
13796:Quasi-experiment
13746:Adaptive designs
13597:
13596:
13584:
13583:
13461:Rank correlation
13223:
13222:
13214:
13213:
13201:
13200:
13168:
13161:
13154:
13145:
13144:
13140:
13133:"Log-likelihood"
13116:
13091:Ward, Michael D.
13086:
13074:
13063:
13044:
13021:
13019:
13009:
12984:
12958:
12932:
12905:
12877:
12876:
12871:(2nd ed.).
12864:
12858:
12857:
12845:
12839:
12838:
12823:
12817:
12816:
12791:
12785:
12778:
12765:
12758:
12745:
12738:
12723:
12716:
12703:
12696:
12683:
12682:
12670:
12664:
12663:
12651:
12645:
12644:
12642:
12620:
12614:
12613:
12585:
12579:
12578:
12566:
12560:
12559:
12541:
12531:
12496:
12490:
12489:
12474:
12468:
12467:
12457:
12428:
12422:
12413:
12407:
12406:
12394:
12388:
12387:
12378:(1â2): 203â207.
12365:
12359:
12358:
12338:
12332:
12331:
12322:(352): 903â904.
12311:
12305:
12304:
12295:(357): 147â148.
12282:
12276:
12275:
12250:
12241:
12240:
12226:
12220:
12219:
12199:
12193:
12192:
12159:
12153:
12152:
12132:
12126:
12125:
12095:
12089:
12088:
12067:
12061:
12060:
12040:
12034:
12033:
12009:
12003:
12002:
11980:
11974:
11973:
11951:
11945:
11944:
11928:
11915:Amemiya, Takeshi
11911:
11905:
11904:
11902:
11901:
11888:Wen Hsiang Wei.
11885:
11879:
11878:
11866:
11860:
11858:
11844:
11835:
11833:
11821:
11810:
11808:
11800:
11794:
11783:
11777:
11770:
11759:
11757:
11734:
11728:
11726:
11715:
11702:
11701:
11679:
11673:
11672:
11654:
11648:
11647:
11627:
11621:
11620:
11600:
11594:
11593:
11575:
11569:
11568:
11538:
11532:
11531:
11503:
11497:
11496:
11486:
11460:
11454:
11453:
11430:
11424:
11423:
11415:
11409:
11408:
11393:
11387:
11386:
11368:
11362:
11361:
11343:
11337:
11336:
11318:
11312:
11311:
11293:
11276:
11270:
11248:Pseudolikelihood
11177:
11174:
11156:
11149:
11047:
11040:
11036:
11033:
11027:
11022:this article by
11013:inline citations
11000:
10999:
10992:
10982:
10980:
10979:
10974:
10962:
10960:
10959:
10954:
10940:
10938:
10937:
10932:
10920:
10918:
10917:
10912:
10852:
10849:
10839:You can help by
10832:
10825:
10783:support function
10779:A. W. F. Edwards
10709:
10707:
10706:
10701:
10698:
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10687:
10682:
10667:
10659:
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10381:
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10320:
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10308:
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10289:
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10264:
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10231:
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10220:
10202:
10201:
10183:
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10180:
10175:
10164:
10156:
10151:
10149:
10141:
10119:
10118:
10102:
10091:
10089:
10088:
10083:
10072:with respect to
10065:
10063:
10062:
10057:
9958:
9957:
9936:
9934:
9933:
9928:
9916:
9914:
9913:
9908:
9894:
9892:
9891:
9886:
9881:
9880:
9865:
9864:
9849:
9847:
9833:
9832:
9823:
9797:
9796:
9781:
9779:
9778:
9773:
9761:
9759:
9758:
9753:
9729:
9723:
9717:
9715:
9713:
9712:
9707:
9702:
9678:
9676:
9674:
9673:
9668:
9657:
9640:
9638:
9636:
9635:
9630:
9628:
9612:
9610:
9609:
9604:
9596:
9570:
9562:
9542:
9518:
9516:
9515:
9510:
9489:
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9486:
9481:
9476:
9468:
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9417:
9391:
9380:
9372:
9352:
9330:
9328:
9327:
9322:
9317:
9316:
9307:
9281:
9270:
9262:
9254:
9253:
9223:
9193:
9191:
9190:
9185:
9124:
9122:
9121:
9116:
9114:
9113:
9105:
9088:
9086:
9085:
9080:
9078:
9077:
9064:
9059:
9058:
9057:
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9051:
9043:
9028:
9026:
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9018:
9007:
9006:
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9000:
8992:
8985:
8984:
8968:
8966:
8965:
8960:
8958:
8954:
8953:
8948:
8947:
8939:
8924:
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8921:
8916:
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8902:
8901:
8896:
8891:
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8874:
8862:
8861:
8856:
8855:
8847:
8836:
8834:
8833:
8828:
8826:
8806:
8804:
8803:
8798:
8795:
8787:
8767:
8765:
8764:
8759:
8739:
8737:
8736:
8731:
8729:
8728:
8723:
8710:
8708:
8707:
8702:
8694:
8693:
8688:
8678:
8670:
8654:inverse function
8651:
8649:
8648:
8643:
8641:
8629:
8627:
8626:
8621:
8619:
8602:
8601:
8562:
8560:
8559:
8554:
8543:
8542:
8533:
8532:
8511:
8510:
8450:A. W. F. Edwards
8410:
8408:
8407:
8402:
8358:
8357:
8333:
8332:
8317:
8315:
8305:
8304:
8297:
8287:
8286:
8279:
8205:
8203:
8202:
8197:
8195:
8194:
8181:
8175:
8173:
8171:
8170:
8165:
8151:
8106:
8104:
8103:
8098:
8090:
8089:
8062:
8061:
8034:
8033:
8021:
8020:
7907:
7905:
7904:
7899:
7897:
7896:
7873:
7871:
7870:
7865:
7863:
7862:
7839:
7837:
7836:
7831:
7829:
7828:
7823:
7806:
7804:
7803:
7798:
7795:
7794:
7788:
7783:
7777:
7776:
7768:
7764:
7763:
7762:
7757:
7750:
7749:
7743:
7738:
7726:
7725:
7720:
7711:
7710:
7705:
7692:
7690:
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7684:
7682:
7677:
7673:
7672:
7671:
7666:
7657:
7646:
7645:
7639:
7634:
7628:
7627:
7619:
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7614:
7613:
7608:
7602:
7598:
7597:
7596:
7591:
7582:
7571:
7570:
7564:
7559:
7544:
7543:
7538:
7537:
7529:
7518:
7516:
7515:
7510:
7508:
7507:
7491:
7489:
7488:
7483:
7481:
7477:
7476:
7475:
7466:
7465:
7460:
7451:
7440:
7439:
7433:
7428:
7422:
7421:
7413:
7409:
7408:
7407:
7402:
7395:
7394:
7388:
7383:
7365:
7364:
7352:
7351:
7335:
7333:
7332:
7327:
7325:
7324:
7308:
7306:
7305:
7300:
7298:
7294:
7293:
7292:
7287:
7278:
7277:
7272:
7258:
7243:
7241:
7240:
7235:
7233:
7229:
7228:
7227:
7215:
7214:
7183:
7181:
7180:
7175:
7164:
7156:
7133:
7131:
7130:
7125:
7123:
7119:
7118:
7113:
7103:
7102:
7097:
7096:
7088:
7081:
7080:
7075:
7074:
7066:
7055:
7053:
7052:
7047:
7045:
7041:
7040:
7039:
7034:
7025:
7024:
7019:
7005:
6993:
6991:
6990:
6985:
6983:
6946:
6934:
6930:
6892:
6888:
6882:
6880:
6879:
6874:
6869:
6865:
6864:
6856:
6819:
6813:
6808:
6792:
6790:
6789:
6784:
6779:
6778:
6770:
6764:
6763:
6750:
6748:
6747:
6742:
6737:
6735:
6725:
6724:
6716:
6710:
6709:
6702:
6686:
6685:
6678:
6654:
6646:
6644:
6643:
6638:
6636:
6635:
6627:
6617:
6613:
6611:
6610:
6605:
6603:
6602:
6594:
6584:
6545:
6543:
6542:
6537:
6523:
6522:
6510:
6509:
6488:
6487:
6475:
6474:
6447:
6446:
6434:
6433:
6407:
6405:
6403:
6402:
6397:
6383:
6381:
6379:
6378:
6373:
6371:
6370:
6352:
6350:
6348:
6347:
6342:
6340:
6339:
6275:, the so-called
6251:
6249:
6248:
6243:
6238:
6236:
6226:
6225:
6213:
6212:
6205:
6195:
6194:
6182:
6181:
6174:
6160:
6159:
6147:
6146:
6123:likelihood ratio
6106:Likelihood ratio
6101:Pseudo-R-squared
6071:
6069:
6068:
6063:
6060:
6056:
6046:
6024:
6022:
6021:
6016:
6011:
6004:
5998:
5997:
5995:
5994:
5993:
5980:
5966:
5962:
5961:
5959:
5958:
5957:
5944:
5930:
5927:
5922:
5898:
5878:
5876:
5875:
5870:
5849:
5835:
5834:
5818:
5813:
5793:
5791:
5790:
5785:
5773:
5771:
5770:
5765:
5754:
5753:
5735:
5731:
5729:
5728:
5727:
5714:
5713:
5700:
5699:
5686:
5682:
5681:
5671:
5651:
5650:
5635:
5631:
5629:
5628:
5627:
5614:
5613:
5600:
5596:
5595:
5585:
5565:
5564:
5552:
5548:
5546:
5545:
5544:
5531:
5523:
5511:it must be that
5510:
5508:
5507:
5502:
5482:
5480:
5479:
5474:
5461:Taylor expansion
5458:
5456:
5455:
5450:
5400:
5398:
5397:
5392:
5389:
5387:
5386:
5385:
5372:
5371:
5358:
5357:
5344:
5334:
5333:
5323:
5316:
5314:
5313:
5312:
5300:
5299:
5286:
5276:
5275:
5265:
5258:
5256:
5255:
5254:
5241:
5227:
5219:
5217:
5216:
5211:
5188:
5186:
5185:
5180:
5149:
5147:
5146:
5141:
5127:
5125:
5124:
5119:
5094:
5069:
5067:
5066:
5061:
5037:
5035:
5034:
5029:
5025:
5024:
5023:
5022:
5011:
5000:
4996:
4994:
4992:
4990:
4989:
4975:
4967:
4941:
4939:
4938:
4933:
4909:
4907:
4906:
4901:
4897:
4896:
4895:
4894:
4881:
4880:
4879:
4868:
4845:
4841:
4839:
4837:
4835:
4834:
4821:
4820:
4806:
4802:
4801:
4791:
4770:
4755:
4753:
4752:
4747:
4739:
4738:
4730:
4720:
4718:
4717:
4712:
4706:
4705:
4700:
4680:
4678:
4677:
4672:
4623:
4621:
4620:
4615:
4603:
4601:
4600:
4595:
4579:
4577:
4576:
4571:
4559:
4557:
4556:
4551:
4539:
4537:
4536:
4531:
4519:
4517:
4516:
4511:
4497:
4496:
4469:
4468:
4454:
4452:
4451:
4446:
4434:
4432:
4431:
4426:
4414:
4412:
4411:
4406:
4379:
4377:
4376:
4371:
4360:
4359:
4333:counting measure
4321:density function
4306:
4304:
4303:
4298:
4296:
4295:
4279:
4277:
4276:
4271:
4269:
4268:
4252:
4250:
4249:
4244:
4230:
4229:
4210:
4205:
4204:
4175:
4174:
4156:
4155:
4145:
4140:
4139:
4107:
4105:
4104:
4099:
4097:
4081:
4080:
4061:
4056:
4055:
4029:
4025:
3998:
3991:
3990:
3980:
3979:
3978:
3964:
3956:
3953:
3952:
3951:
3922:
3917:
3916:
3891:
3882:
3878:
3865:
3864:
3852:
3851:
3824:
3823:
3816:
3815:
3814:
3785:
3780:
3779:
3750:
3749:
3731:
3730:
3720:
3715:
3714:
3690:
3677:
3675:
3674:
3669:
3655:
3654:
3610:
3603:
3602:
3592:
3591:
3590:
3576:
3568:
3565:
3564:
3563:
3530:
3528:
3527:
3522:
3491:
3484:
3483:
3473:
3472:
3471:
3457:
3449:
3443:
3438:
3437:
3399:
3398:
3386:
3385:
3358:
3357:
3347:
3342:
3341:
3309:
3307:
3306:
3301:
3272:
3270:
3269:
3264:
3233:
3226:
3225:
3215:
3214:
3213:
3199:
3191:
3185:
3180:
3179:
3138:
3137:
3119:
3118:
3103:
3095:
3089:
3084:
3083:
3045:
3044:
3032:
3031:
3004:
3003:
2997:
2989:
2983:
2978:
2977:
2947:
2945:
2944:
2939:
2927:
2925:
2924:
2919:
2902:
2901:
2889:
2888:
2861:
2860:
2854:
2846:
2840:
2835:
2834:
2796:
2795:
2783:
2782:
2755:
2754:
2744:
2739:
2738:
2708:
2706:
2705:
2700:
2686:
2685:
2673:
2672:
2645:
2644:
2631:
2629:
2628:
2623:
2605:
2603:
2602:
2597:
2586:
2585:
2573:
2572:
2553:
2551:
2550:
2545:
2543:
2542:
2514:
2512:
2511:
2508:{\textstyle X=x}
2506:
2488:
2486:
2485:
2480:
2468:
2466:
2465:
2460:
2448:
2446:
2445:
2440:
2438:
2437:
2424:
2422:
2421:
2418:{\textstyle X=x}
2416:
2395:
2393:
2392:
2387:
2371:
2369:
2368:
2363:
2349:
2347:
2346:
2341:
2327:
2326:
2299:
2298:
2283:
2281:
2280:
2275:
2263:
2261:
2260:
2255:
2243:
2241:
2240:
2235:
2223:density function
2212:
2210:
2209:
2204:
2184:
2182:
2181:
2176:
2174:
2173:
2160:
2158:
2157:
2152:
2150:
2149:
2146:
2131:
2129:
2128:
2123:
2112:
2109:
2098:
2097:
2094:
2085:
2084:
2067:
2065:
2064:
2059:
2051:
2050:
2029:
2028:
2025:
2016:
2013:
1996:
1994:
1993:
1988:
1980:
1979:
1976:
1960:
1958:
1957:
1952:
1947:
1944:
1929:
1927:
1926:
1921:
1910:
1909:
1906:
1883:
1881:
1880:
1875:
1849:
1848:
1845:
1824:
1822:
1821:
1816:
1805:
1802:
1791:
1790:
1787:
1778:
1777:
1762:
1760:
1759:
1754:
1746:
1745:
1742:
1727:
1725:
1724:
1719:
1711:
1710:
1689:
1688:
1685:
1676:
1673:
1649:
1647:
1646:
1641:
1633:
1632:
1629:
1612:
1610:
1609:
1604:
1602:
1601:
1598:
1585:
1583:
1582:
1577:
1575:
1574:
1571:
1554:
1552:
1551:
1546:
1541:
1540:
1537:
1521:
1516:
1513:
1492:
1490:
1489:
1484:
1481:
1476:
1473:
1448:
1446:
1445:
1440:
1428:
1426:
1425:
1420:
1408:
1406:
1405:
1400:
1373:
1371:
1370:
1365:
1348:
1347:
1334:
1332:
1331:
1326:
1310:
1308:
1307:
1302:
1290:
1288:
1287:
1282:
1270:
1268:
1267:
1262:
1250:
1231:
1213:" is written as
1212:
1209:
1207:
1206:
1201:
1189:
1187:
1186:
1181:
1169:
1167:
1166:
1161:
1149:
1147:
1146:
1141:
1129:
1127:
1126:
1121:
1102:
1100:
1099:
1094:
1080:
1078:
1077:
1072:
1052:
1051:
1030:
1029:
1002:
1001:
986:
984:
983:
978:
966:
964:
963:
958:
939:
937:
936:
931:
907:
905:
904:
901:{\textstyle X=x}
899:
881:
879:
878:
873:
850:
848:
847:
842:
830:
828:
827:
822:
795:
793:
792:
787:
760:
758:
757:
752:
721:
719:
718:
713:
701:
699:
698:
693:
681:
679:
678:
673:
661:
659:
658:
653:
641:
639:
638:
633:
604:
602:
601:
596:
576:
575:
562:
560:
559:
554:
518:
516:
515:
510:
498:
496:
495:
490:
476:
474:
473:
468:
423:
421:
420:
415:
380:In contrast, in
312:
305:
298:
282:
281:
248:Model evaluation
49:
30:
29:
21:
15522:
15521:
15517:
15516:
15515:
15513:
15512:
15511:
15492:
15491:
15490:
15478:
15470:
15468:
15463:
15426:
15397:
15359:
15296:
15282:quality control
15249:
15231:Clinical trials
15208:
15183:
15167:
15155:Hazard function
15149:
15103:
15065:
15049:
15012:
15008:BreuschâGodfrey
14996:
14973:
14913:
14888:Factor analysis
14834:
14815:Graphical model
14787:
14754:
14721:
14707:
14687:
14641:
14608:
14570:
14533:
14532:
14501:
14445:
14432:
14424:
14416:
14400:
14385:
14364:Rank statistics
14358:
14337:Model selection
14325:
14283:Goodness of fit
14277:
14254:
14228:
14200:
14153:
14098:
14087:Median unbiased
14015:
13926:
13859:Order statistic
13821:
13800:
13767:
13741:
13693:
13648:
13591:
13589:Data collection
13570:
13482:
13437:
13411:
13389:
13349:
13301:
13218:Continuous data
13208:
13195:
13177:
13172:
13131:
13123:
13113:
13083:
13060:
13041:
12981:
12955:
12929:
12902:
12886:
12884:Further reading
12881:
12880:
12875:. chap. 7.
12873:Springer-Verlag
12865:
12861:
12846:
12842:
12831:Fienberg, S. E.
12824:
12820:
12814:
12792:
12788:
12779:
12768:
12760:D. V. Lindley:
12759:
12748:
12739:
12726:
12717:
12706:
12697:
12686:
12671:
12667:
12652:
12648:
12621:
12617:
12586:
12582:
12567:
12563:
12497:
12493:
12475:
12471:
12429:
12425:
12414:
12410:
12395:
12391:
12366:
12362:
12339:
12335:
12312:
12308:
12283:
12279:
12272:
12251:
12244:
12227:
12223:
12216:
12200:
12196:
12160:
12156:
12133:
12129:
12114:10.2307/2347496
12096:
12092:
12085:
12068:
12064:
12057:
12041:
12037:
12030:
12010:
12006:
11999:
11981:
11977:
11970:
11952:
11948:
11941:
11912:
11908:
11899:
11897:
11886:
11882:
11867:
11863:
11845:
11838:
11822:
11813:
11801:
11797:
11784:
11780:
11771:
11762:
11755:
11735:
11731:
11716:
11705:
11692:(3a): 153â157.
11680:
11676:
11669:
11655:
11651:
11628:
11624:
11601:
11597:
11590:
11576:
11572:
11557:10.2307/2333005
11539:
11535:
11504:
11500:
11461:
11457:
11450:
11431:
11427:
11416:
11412:
11394:
11390:
11383:
11369:
11365:
11358:
11344:
11340:
11333:
11319:
11315:
11308:
11294:
11290:
11285:
11280:
11279:
11271:
11267:
11262:
11257:
11198:
11178:
11172:
11169:
11162:needs expansion
11147:
11127:
11120:
11071:
11064:
11048:
11037:
11031:
11028:
11018:Please help to
11017:
11001:
10997:
10990:
10968:
10965:
10964:
10948:
10945:
10944:
10943:probability of
10926:
10923:
10922:
10906:
10903:
10902:
10870:random variable
10862:
10853:
10847:
10844:
10823:
10795:
10735:
10725:
10720:
10693:
10689:
10683:
10672:
10658:
10644:
10643:
10640:
10637:
10636:
10614:
10613:
10611:
10608:
10607:
10579:
10574:
10560:
10559:
10557:
10554:
10553:
10535:
10532:
10531:
10515:
10514:
10505:
10501:
10495:
10484:
10464:
10462:
10449:
10439:
10435:
10414:
10413:
10403:
10401:
10382:
10372:
10368:
10347:
10346:
10336:
10334:
10332:
10330:
10324:
10323:
10313:
10303:
10299:
10284:
10280:
10259:
10258:
10248:
10246:
10241:
10239:
10236:
10235:
10216:
10212:
10197:
10193:
10191:
10188:
10187:
10155:
10142:
10114:
10113:
10103:
10101:
10099:
10096:
10095:
10077:
10074:
10073:
9953:
9952:
9944:
9941:
9940:
9922:
9919:
9918:
9902:
9899:
9898:
9870:
9866:
9854:
9850:
9834:
9828:
9824:
9822:
9792:
9791:
9789:
9786:
9785:
9767:
9764:
9763:
9747:
9744:
9743:
9736:
9725:
9719:
9698:
9690:
9687:
9686:
9684:
9653:
9651:
9648:
9647:
9645:
9624:
9622:
9619:
9618:
9616:
9592:
9566:
9558:
9538:
9530:
9527:
9526:
9495:
9492:
9491:
9472:
9464:
9462:
9459:
9458:
9413:
9387:
9376:
9368:
9348:
9340:
9337:
9336:
9312:
9311:
9303:
9277:
9266:
9258:
9249:
9248:
9219:
9205:
9202:
9201:
9167:
9164:
9163:
9145:
9139:
9104:
9103:
9101:
9098:
9097:
9091:Rolle's theorem
9073:
9069:
9053:
9042:
9041:
9040:
9038:
9035:
9034:
9029:asymptotically
9014:
9002:
8991:
8990:
8989:
8980:
8976:
8974:
8971:
8970:
8949:
8938:
8937:
8936:
8932:
8930:
8927:
8926:
8910:
8907:
8906:
8887:
8875:
8870:
8857:
8846:
8845:
8844:
8842:
8839:
8838:
8822:
8820:
8817:
8816:
8788:
8783:
8777:
8774:
8773:
8753:
8750:
8749:
8746:Euclidean space
8743:
8724:
8719:
8718:
8716:
8713:
8712:
8689:
8684:
8683:
8671:
8666:
8660:
8657:
8656:
8637:
8635:
8632:
8631:
8615:
8597:
8593:
8591:
8588:
8587:
8538:
8534:
8528:
8524:
8506:
8502:
8500:
8497:
8496:
8481:
8439:parameter space
8435:support surface
8419:
8353:
8352:
8328:
8327:
8300:
8299:
8298:
8282:
8281:
8280:
8278:
8270:
8267:
8266:
8243:log-probability
8190:
8189:
8187:
8184:
8183:
8177:
8159:
8156:
8155:
8153:
8147:
8141:
8139:Log-probability
8135:
8085:
8081:
8057:
8053:
8029:
8025:
8016:
8012:
7998:
7995:
7994:
7984:
7972:
7955:
7949:
7926:
7918:standard errors
7892:
7888:
7886:
7883:
7882:
7858:
7854:
7852:
7849:
7848:
7824:
7819:
7818:
7816:
7813:
7812:
7790:
7789:
7784:
7779:
7769:
7758:
7753:
7752:
7745:
7744:
7739:
7734:
7732:
7728:
7727:
7721:
7716:
7715:
7706:
7701:
7700:
7698:
7695:
7694:
7678:
7667:
7662:
7661:
7653:
7652:
7648:
7641:
7640:
7635:
7630:
7620:
7609:
7604:
7603:
7592:
7587:
7586:
7578:
7577:
7573:
7566:
7565:
7560:
7555:
7553:
7549:
7548:
7539:
7528:
7527:
7526:
7524:
7521:
7520:
7503:
7499:
7497:
7494:
7493:
7471:
7467:
7461:
7456:
7455:
7447:
7446:
7442:
7435:
7434:
7429:
7424:
7414:
7403:
7398:
7397:
7390:
7389:
7384:
7379:
7377:
7373:
7372:
7360:
7356:
7347:
7343:
7341:
7338:
7337:
7320:
7316:
7314:
7311:
7310:
7288:
7283:
7282:
7273:
7268:
7267:
7266:
7262:
7254:
7252:
7249:
7248:
7223:
7219:
7210:
7206:
7205:
7201:
7193:
7190:
7189:
7160:
7152:
7150:
7147:
7146:
7114:
7109:
7108:
7104:
7098:
7087:
7086:
7085:
7076:
7065:
7064:
7063:
7061:
7058:
7057:
7035:
7030:
7029:
7020:
7015:
7014:
7013:
7009:
7001:
6999:
6996:
6995:
6979:
6977:
6974:
6973:
6969:
6953:
6944:
6932:
6928:
6924:(Bayesianism).
6890:
6886:
6855:
6833:
6829:
6827:
6824:
6823:
6817:
6811:
6806:
6799:
6769:
6768:
6759:
6758:
6756:
6753:
6752:
6715:
6714:
6705:
6704:
6703:
6681:
6680:
6679:
6677:
6660:
6657:
6656:
6652:
6626:
6625:
6623:
6620:
6619:
6615:
6593:
6592:
6590:
6587:
6586:
6582:
6575:
6569:
6561:diagnostic test
6518:
6514:
6505:
6501:
6483:
6479:
6470:
6466:
6442:
6438:
6429:
6425:
6417:
6414:
6413:
6391:
6388:
6387:
6385:
6366:
6362:
6360:
6357:
6356:
6354:
6335:
6331:
6329:
6326:
6325:
6323:
6221:
6217:
6208:
6207:
6206:
6190:
6186:
6177:
6176:
6175:
6173:
6155:
6151:
6142:
6138:
6130:
6127:
6126:
6119:
6108:
6103:
6097:
6042:
6041:
6037:
6034:
6031:
6030:
6007:
5989:
5985:
5981:
5967:
5965:
5953:
5949:
5945:
5931:
5929:
5923:
5915:
5894:
5892:
5889:
5888:
5845:
5824:
5820:
5814:
5806:
5799:
5796:
5795:
5779:
5776:
5775:
5743:
5739:
5723:
5719:
5709:
5705:
5695:
5691:
5687:
5677:
5673:
5672:
5670:
5666:
5643:
5639:
5623:
5619:
5609:
5605:
5601:
5591:
5587:
5586:
5584:
5580:
5560:
5556:
5540:
5536:
5532:
5524:
5522:
5518:
5516:
5513:
5512:
5488:
5485:
5484:
5468:
5465:
5464:
5406:
5403:
5402:
5381:
5377:
5367:
5363:
5353:
5349:
5345:
5329:
5325:
5324:
5322:
5308:
5304:
5295:
5291:
5287:
5271:
5267:
5266:
5264:
5250:
5246:
5242:
5228:
5226:
5224:
5221:
5220:
5194:
5191:
5190:
5174:
5171:
5170:
5133:
5130:
5129:
5081:
5075:
5072:
5071:
5047:
5044:
5043:
5018:
5017:
5013:
5012:
5001:
4985:
4981:
4976:
4968:
4966:
4964:
4960:
4947:
4944:
4943:
4919:
4916:
4915:
4890:
4889:
4885:
4875:
4874:
4870:
4869:
4846:
4830:
4826:
4816:
4812:
4807:
4797:
4793:
4792:
4790:
4788:
4784:
4766:
4764:
4761:
4760:
4729:
4728:
4726:
4723:
4722:
4701:
4696:
4695:
4693:
4690:
4689:
4666:
4663:
4662:
4660:
4630:
4609:
4606:
4605:
4589:
4586:
4585:
4565:
4562:
4561:
4545:
4542:
4541:
4525:
4522:
4521:
4492:
4488:
4464:
4463:
4461:
4458:
4457:
4440:
4437:
4436:
4420:
4417:
4416:
4385:
4382:
4381:
4355:
4351:
4349:
4346:
4345:
4341:
4313:
4291:
4287:
4285:
4282:
4281:
4264:
4260:
4258:
4255:
4254:
4225:
4221:
4206:
4184:
4183:
4170:
4166:
4151:
4150:
4141:
4119:
4118:
4115:
4112:
4111:
4095:
4094:
4076:
4072:
4057:
4035:
4034:
3986:
3982:
3981:
3974:
3970:
3969:
3955:
3947:
3943:
3936:
3931:
3927:
3918:
3896:
3895:
3892:
3890:
3884:
3883:
3860:
3856:
3847:
3843:
3819:
3818:
3810:
3806:
3799:
3794:
3790:
3781:
3759:
3758:
3745:
3741:
3726:
3725:
3716:
3694:
3693:
3687:
3685:
3682:
3681:
3650:
3646:
3598:
3594:
3593:
3586:
3582:
3581:
3567:
3559:
3555:
3548:
3542:
3539:
3538:
3479:
3475:
3474:
3467:
3463:
3462:
3448:
3439:
3417:
3416:
3394:
3390:
3381:
3377:
3353:
3352:
3343:
3321:
3320:
3317:
3314:
3313:
3280:
3277:
3276:
3221:
3217:
3216:
3209:
3205:
3204:
3190:
3181:
3159:
3158:
3133:
3129:
3114:
3110:
3094:
3085:
3063:
3062:
3040:
3036:
3027:
3023:
2999:
2998:
2988:
2979:
2957:
2956:
2953:
2950:
2949:
2933:
2930:
2929:
2897:
2893:
2884:
2880:
2856:
2855:
2845:
2836:
2814:
2813:
2791:
2787:
2778:
2774:
2750:
2749:
2740:
2718:
2717:
2714:
2711:
2710:
2709:. Observe that
2681:
2677:
2668:
2664:
2640:
2639:
2637:
2634:
2633:
2611:
2608:
2607:
2581:
2577:
2568:
2564:
2559:
2556:
2555:
2538:
2534:
2532:
2529:
2528:
2523:The use of the
2521:
2494:
2491:
2490:
2474:
2471:
2470:
2454:
2451:
2450:
2433:
2432:
2430:
2427:
2426:
2404:
2401:
2400:
2381:
2378:
2377:
2357:
2354:
2353:
2322:
2318:
2294:
2293:
2291:
2288:
2287:
2269:
2266:
2265:
2249:
2246:
2245:
2244:(a function of
2229:
2226:
2225:
2215:random variable
2198:
2195:
2194:
2191:
2169:
2168:
2166:
2163:
2162:
2145:
2141:
2139:
2136:
2135:
2108:
2093:
2089:
2080:
2079:
2077:
2074:
2073:
2046:
2042:
2024:
2020:
2012:
2004:
2001:
2000:
1975:
1971:
1969:
1966:
1965:
1943:
1935:
1932:
1931:
1905:
1901:
1893:
1890:
1889:
1844:
1840:
1832:
1829:
1828:
1801:
1786:
1782:
1773:
1772:
1770:
1767:
1766:
1741:
1737:
1735:
1732:
1731:
1706:
1702:
1684:
1680:
1672:
1664:
1661:
1660:
1628:
1624:
1622:
1619:
1618:
1597:
1593:
1591:
1588:
1587:
1570:
1566:
1564:
1561:
1560:
1536:
1532:
1517:
1512:
1506:
1503:
1502:
1477:
1472:
1466:
1463:
1462:
1455:
1434:
1431:
1430:
1429:given the data
1414:
1411:
1410:
1379:
1376:
1375:
1343:
1342:
1340:
1337:
1336:
1320:
1317:
1316:
1296:
1293:
1292:
1276:
1273:
1272:
1256:
1253:
1252:
1233:
1214:
1210:
1195:
1192:
1191:
1175:
1172:
1171:
1155:
1152:
1151:
1135:
1132:
1131:
1115:
1112:
1111:
1088:
1085:
1084:
1047:
1043:
1025:
1021:
997:
996:
994:
991:
990:
972:
969:
968:
952:
949:
948:
942:random variable
925:
922:
921:
918:
887:
884:
883:
867:
864:
863:
853:random variable
836:
833:
832:
801:
798:
797:
766:
763:
762:
731:
728:
727:
726:, the notation
707:
704:
703:
687:
684:
683:
667:
664:
663:
647:
644:
643:
612:
609:
608:
571:
570:
568:
565:
564:
524:
521:
520:
504:
501:
500:
484:
481:
480:
438:
435:
434:
409:
406:
405:
402:
348:random variable
316:
276:
261:Model averaging
240:Nested sampling
152:Empirical Bayes
142:Conjugate prior
111:Cromwell's rule
28:
23:
22:
15:
12:
11:
5:
15520:
15510:
15509:
15504:
15489:
15488:
15465:
15464:
15462:
15461:
15449:
15437:
15423:
15410:
15407:
15406:
15403:
15402:
15399:
15398:
15396:
15395:
15390:
15385:
15380:
15375:
15369:
15367:
15361:
15360:
15358:
15357:
15352:
15347:
15342:
15337:
15332:
15327:
15322:
15317:
15312:
15306:
15304:
15298:
15297:
15295:
15294:
15289:
15284:
15275:
15270:
15265:
15259:
15257:
15251:
15250:
15248:
15247:
15242:
15237:
15228:
15226:Bioinformatics
15222:
15220:
15210:
15209:
15197:
15196:
15193:
15192:
15189:
15188:
15185:
15184:
15182:
15181:
15175:
15173:
15169:
15168:
15166:
15165:
15159:
15157:
15151:
15150:
15148:
15147:
15142:
15137:
15132:
15126:
15124:
15115:
15109:
15108:
15105:
15104:
15102:
15101:
15096:
15091:
15086:
15081:
15075:
15073:
15067:
15066:
15064:
15063:
15058:
15053:
15045:
15040:
15035:
15034:
15033:
15031:partial (PACF)
15022:
15020:
15014:
15013:
15011:
15010:
15005:
15000:
14992:
14987:
14981:
14979:
14978:Specific tests
14975:
14974:
14972:
14971:
14966:
14961:
14956:
14951:
14946:
14941:
14936:
14930:
14928:
14921:
14915:
14914:
14912:
14911:
14910:
14909:
14908:
14907:
14892:
14891:
14890:
14880:
14878:Classification
14875:
14870:
14865:
14860:
14855:
14850:
14844:
14842:
14836:
14835:
14833:
14832:
14827:
14825:McNemar's test
14822:
14817:
14812:
14807:
14801:
14799:
14789:
14788:
14764:
14763:
14760:
14759:
14756:
14755:
14753:
14752:
14747:
14742:
14737:
14731:
14729:
14723:
14722:
14720:
14719:
14703:
14697:
14695:
14689:
14688:
14686:
14685:
14680:
14675:
14670:
14665:
14663:Semiparametric
14660:
14655:
14649:
14647:
14643:
14642:
14640:
14639:
14634:
14629:
14624:
14618:
14616:
14610:
14609:
14607:
14606:
14601:
14596:
14591:
14586:
14580:
14578:
14572:
14571:
14569:
14568:
14563:
14558:
14553:
14547:
14545:
14535:
14534:
14531:
14530:
14525:
14519:
14511:
14510:
14507:
14506:
14503:
14502:
14500:
14499:
14498:
14497:
14487:
14482:
14477:
14476:
14475:
14470:
14459:
14457:
14451:
14450:
14447:
14446:
14444:
14443:
14438:
14437:
14436:
14428:
14420:
14404:
14401:(MannâWhitney)
14396:
14395:
14394:
14381:
14380:
14379:
14368:
14366:
14360:
14359:
14357:
14356:
14355:
14354:
14349:
14344:
14334:
14329:
14326:(ShapiroâWilk)
14321:
14316:
14311:
14306:
14301:
14293:
14287:
14285:
14279:
14278:
14276:
14275:
14267:
14258:
14246:
14240:
14238:Specific tests
14234:
14233:
14230:
14229:
14227:
14226:
14221:
14216:
14210:
14208:
14202:
14201:
14199:
14198:
14193:
14192:
14191:
14181:
14180:
14179:
14169:
14163:
14161:
14155:
14154:
14152:
14151:
14150:
14149:
14144:
14134:
14129:
14124:
14119:
14114:
14108:
14106:
14100:
14099:
14097:
14096:
14091:
14090:
14089:
14084:
14083:
14082:
14077:
14062:
14061:
14060:
14055:
14050:
14045:
14034:
14032:
14023:
14017:
14016:
14014:
14013:
14008:
14003:
14002:
14001:
13991:
13986:
13985:
13984:
13974:
13973:
13972:
13967:
13962:
13952:
13947:
13942:
13941:
13940:
13935:
13930:
13914:
13913:
13912:
13907:
13902:
13892:
13891:
13890:
13885:
13875:
13874:
13873:
13863:
13862:
13861:
13851:
13846:
13841:
13835:
13833:
13823:
13822:
13810:
13809:
13806:
13805:
13802:
13801:
13799:
13798:
13793:
13788:
13783:
13777:
13775:
13769:
13768:
13766:
13765:
13760:
13755:
13749:
13747:
13743:
13742:
13740:
13739:
13734:
13729:
13724:
13719:
13714:
13709:
13703:
13701:
13695:
13694:
13692:
13691:
13689:Standard error
13686:
13681:
13676:
13675:
13674:
13669:
13658:
13656:
13650:
13649:
13647:
13646:
13641:
13636:
13631:
13626:
13621:
13619:Optimal design
13616:
13611:
13605:
13603:
13593:
13592:
13580:
13579:
13576:
13575:
13572:
13571:
13569:
13568:
13563:
13558:
13553:
13548:
13543:
13538:
13533:
13528:
13523:
13518:
13513:
13508:
13503:
13498:
13492:
13490:
13484:
13483:
13481:
13480:
13475:
13474:
13473:
13468:
13458:
13453:
13447:
13445:
13439:
13438:
13436:
13435:
13430:
13425:
13419:
13417:
13416:Summary tables
13413:
13412:
13410:
13409:
13403:
13401:
13395:
13394:
13391:
13390:
13388:
13387:
13386:
13385:
13380:
13375:
13365:
13359:
13357:
13351:
13350:
13348:
13347:
13342:
13337:
13332:
13327:
13322:
13317:
13311:
13309:
13303:
13302:
13300:
13299:
13294:
13289:
13288:
13287:
13282:
13277:
13272:
13267:
13262:
13257:
13252:
13250:Contraharmonic
13247:
13242:
13231:
13229:
13220:
13210:
13209:
13197:
13196:
13194:
13193:
13188:
13182:
13179:
13178:
13171:
13170:
13163:
13156:
13148:
13142:
13141:
13129:
13122:
13121:External links
13119:
13118:
13117:
13111:
13087:
13081:
13064:
13058:
13045:
13039:
13022:
12985:
12979:
12959:
12953:
12933:
12927:
12906:
12900:
12885:
12882:
12879:
12878:
12859:
12856:. Part I.
12840:
12818:
12812:
12786:
12766:
12746:
12740:E. T. Jaynes:
12724:
12704:
12684:
12665:
12646:
12615:
12596:(4): 528â535.
12580:
12577:. p. 329.
12561:
12491:
12469:
12448:(2): 214â222.
12423:
12415:"likelihood",
12408:
12389:
12360:
12333:
12306:
12277:
12270:
12242:
12236:Stack Exchange
12221:
12214:
12194:
12175:(2): 269â276.
12154:
12143:(3): 311â328.
12127:
12090:
12083:
12071:Aitkin, Murray
12062:
12055:
12035:
12028:
12004:
11997:
11975:
11968:
11946:
11939:
11906:
11880:
11861:
11836:
11811:
11795:
11778:
11760:
11753:
11729:
11703:
11674:
11667:
11649:
11638:(3): 540â546.
11622:
11611:(2): 184â189.
11595:
11588:
11570:
11551:(1â2): 56â61.
11533:
11498:
11477:(4): 758â767.
11455:
11448:
11425:
11410:
11388:
11381:
11363:
11356:
11338:
11331:
11313:
11306:
11287:
11286:
11284:
11281:
11278:
11277:
11264:
11263:
11261:
11258:
11256:
11255:
11250:
11245:
11240:
11235:
11230:
11225:
11220:
11215:
11210:
11205:
11199:
11197:
11194:
11180:
11179:
11159:
11157:
11146:
11143:
11134:Wilks' theorem
11130:Ď distribution
11125:
11118:
11099:Wilks' theorem
11069:
11062:
11050:
11049:
11004:
11002:
10995:
10989:
10986:
10976:{\textstyle X}
10972:
10956:{\textstyle Y}
10952:
10934:{\textstyle Y}
10930:
10914:{\textstyle X}
10910:
10861:
10858:
10855:
10854:
10835:
10833:
10822:
10819:
10794:
10791:
10739:Middle English
10724:
10721:
10719:
10716:
10696:
10692:
10686:
10681:
10678:
10675:
10671:
10665:
10662:
10657:
10651:
10648:
10621:
10618:
10593:
10586:
10583:
10578:
10573:
10567:
10564:
10539:
10513:
10508:
10504:
10498:
10493:
10490:
10487:
10483:
10479:
10474:
10470:
10467:
10461:
10455:
10452:
10447:
10442:
10438:
10434:
10431:
10428:
10425:
10422:
10417:
10412:
10409:
10406:
10400:
10397:
10394:
10388:
10385:
10380:
10375:
10371:
10367:
10364:
10361:
10358:
10355:
10350:
10345:
10342:
10339:
10333:
10329:
10326:
10325:
10319:
10316:
10311:
10306:
10302:
10298:
10295:
10292:
10287:
10283:
10279:
10276:
10273:
10270:
10267:
10262:
10257:
10254:
10251:
10245:
10243:
10219:
10215:
10211:
10208:
10205:
10200:
10196:
10173:
10170:
10167:
10162:
10159:
10154:
10148:
10145:
10140:
10137:
10134:
10131:
10128:
10125:
10122:
10117:
10112:
10109:
10106:
10081:
10054:
10051:
10048:
10045:
10042:
10039:
10036:
10033:
10030:
10027:
10024:
10021:
10018:
10015:
10012:
10009:
10006:
10003:
10000:
9997:
9994:
9991:
9988:
9985:
9982:
9979:
9976:
9973:
9970:
9967:
9964:
9961:
9956:
9951:
9948:
9930:{\textstyle x}
9926:
9906:
9884:
9879:
9876:
9873:
9869:
9863:
9860:
9857:
9853:
9846:
9843:
9840:
9837:
9831:
9827:
9821:
9818:
9815:
9812:
9809:
9806:
9803:
9800:
9795:
9771:
9751:
9735:
9732:
9705:
9701:
9697:
9694:
9666:
9663:
9660:
9656:
9627:
9602:
9599:
9595:
9591:
9588:
9585:
9582:
9579:
9576:
9573:
9569:
9565:
9561:
9557:
9554:
9551:
9548:
9545:
9541:
9537:
9534:
9508:
9505:
9502:
9499:
9479:
9475:
9471:
9467:
9444:
9441:
9438:
9435:
9432:
9429:
9426:
9423:
9420:
9416:
9412:
9409:
9406:
9403:
9400:
9397:
9394:
9390:
9386:
9383:
9379:
9375:
9371:
9367:
9364:
9361:
9358:
9355:
9351:
9347:
9344:
9320:
9315:
9310:
9306:
9302:
9299:
9296:
9293:
9290:
9287:
9284:
9280:
9276:
9273:
9269:
9265:
9261:
9257:
9252:
9247:
9244:
9241:
9238:
9235:
9232:
9229:
9226:
9222:
9218:
9215:
9212:
9209:
9183:
9180:
9177:
9174:
9171:
9157:exponentiation
9138:
9135:
9111:
9108:
9076:
9072:
9062:
9056:
9049:
9046:
9017:
9013:
9010:
9005:
8998:
8995:
8988:
8983:
8979:
8957:
8952:
8945:
8942:
8935:
8914:
8894:
8890:
8886:
8881:
8878:
8873:
8869:
8865:
8860:
8853:
8850:
8825:
8794:
8791:
8786:
8782:
8757:
8741:
8727:
8722:
8700:
8697:
8692:
8687:
8682:
8677:
8674:
8669:
8665:
8640:
8618:
8614:
8611:
8608:
8605:
8600:
8596:
8552:
8549:
8546:
8541:
8537:
8531:
8527:
8523:
8520:
8517:
8514:
8509:
8505:
8480:
8477:
8418:
8415:
8400:
8397:
8394:
8391:
8388:
8385:
8382:
8379:
8376:
8373:
8370:
8367:
8364:
8361:
8356:
8351:
8348:
8345:
8342:
8339:
8336:
8331:
8326:
8323:
8320:
8314:
8311:
8308:
8303:
8296:
8293:
8290:
8285:
8277:
8274:
8193:
8163:
8134:
8133:Log-likelihood
8131:
8127:improper prior
8096:
8093:
8088:
8084:
8080:
8077:
8074:
8071:
8068:
8065:
8060:
8056:
8052:
8049:
8046:
8043:
8040:
8037:
8032:
8028:
8024:
8019:
8015:
8011:
8008:
8005:
8002:
7983:
7980:
7971:
7968:
7951:Main article:
7948:
7945:
7925:
7922:
7895:
7891:
7861:
7857:
7827:
7822:
7793:
7787:
7782:
7775:
7772:
7767:
7761:
7756:
7748:
7742:
7737:
7731:
7724:
7719:
7714:
7709:
7704:
7681:
7676:
7670:
7665:
7660:
7656:
7651:
7644:
7638:
7633:
7626:
7623:
7618:
7612:
7607:
7601:
7595:
7590:
7585:
7581:
7576:
7569:
7563:
7558:
7552:
7547:
7542:
7535:
7532:
7506:
7502:
7480:
7474:
7470:
7464:
7459:
7454:
7450:
7445:
7438:
7432:
7427:
7420:
7417:
7412:
7406:
7401:
7393:
7387:
7382:
7376:
7371:
7368:
7363:
7359:
7355:
7350:
7346:
7323:
7319:
7297:
7291:
7286:
7281:
7276:
7271:
7265:
7261:
7257:
7232:
7226:
7222:
7218:
7213:
7209:
7204:
7200:
7197:
7173:
7170:
7167:
7163:
7159:
7155:
7122:
7117:
7112:
7107:
7101:
7094:
7091:
7084:
7079:
7072:
7069:
7044:
7038:
7033:
7028:
7023:
7018:
7012:
7008:
7004:
6982:
6968:
6965:
6952:
6949:
6937:Wilks' theorem
6872:
6868:
6862:
6859:
6854:
6851:
6848:
6845:
6842:
6839:
6836:
6832:
6798:
6795:
6782:
6776:
6773:
6767:
6762:
6740:
6734:
6731:
6728:
6722:
6719:
6713:
6708:
6701:
6698:
6695:
6692:
6689:
6684:
6676:
6673:
6670:
6667:
6664:
6633:
6630:
6600:
6597:
6568:
6565:
6535:
6532:
6529:
6526:
6521:
6517:
6513:
6508:
6504:
6500:
6497:
6494:
6491:
6486:
6482:
6478:
6473:
6469:
6465:
6462:
6459:
6456:
6453:
6450:
6445:
6441:
6437:
6432:
6428:
6424:
6421:
6395:
6369:
6365:
6338:
6334:
6297:Wilks' theorem
6273:test statistic
6241:
6235:
6232:
6229:
6224:
6220:
6216:
6211:
6204:
6201:
6198:
6193:
6189:
6185:
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6172:
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6163:
6158:
6154:
6150:
6145:
6141:
6137:
6134:
6107:
6104:
6096:
6093:
6059:
6055:
6052:
6049:
6045:
6040:
6014:
6010:
6003:
5992:
5988:
5984:
5979:
5976:
5973:
5970:
5956:
5952:
5948:
5943:
5940:
5937:
5934:
5926:
5921:
5918:
5914:
5910:
5907:
5904:
5901:
5897:
5868:
5864:
5861:
5858:
5855:
5852:
5848:
5844:
5841:
5838:
5833:
5830:
5827:
5823:
5817:
5812:
5809:
5805:
5787:{\textstyle H}
5783:
5763:
5760:
5757:
5752:
5749:
5746:
5742:
5738:
5734:
5726:
5722:
5718:
5712:
5708:
5704:
5698:
5694:
5690:
5685:
5680:
5676:
5669:
5664:
5660:
5657:
5654:
5649:
5646:
5642:
5638:
5634:
5626:
5622:
5618:
5612:
5608:
5604:
5599:
5594:
5590:
5583:
5578:
5574:
5571:
5568:
5563:
5559:
5555:
5551:
5543:
5539:
5535:
5530:
5527:
5521:
5499:
5496:
5493:
5483:and for every
5476:{\textstyle x}
5472:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5401:exist for all
5384:
5380:
5376:
5370:
5366:
5362:
5356:
5352:
5348:
5343:
5340:
5337:
5332:
5328:
5320:
5311:
5307:
5303:
5298:
5294:
5290:
5285:
5282:
5279:
5274:
5270:
5262:
5253:
5249:
5245:
5240:
5237:
5234:
5231:
5209:
5205:
5202:
5199:
5189:, and for all
5182:{\textstyle x}
5178:
5138:
5117:
5113:
5110:
5107:
5104:
5101:
5098:
5093:
5090:
5087:
5084:
5080:
5059:
5055:
5052:
5021:
5016:
5010:
5007:
5004:
4999:
4988:
4984:
4980:
4974:
4971:
4963:
4958:
4955:
4952:
4930:
4927:
4924:
4893:
4888:
4884:
4878:
4873:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4844:
4833:
4829:
4825:
4819:
4815:
4811:
4805:
4800:
4796:
4787:
4782:
4779:
4776:
4773:
4769:
4745:
4742:
4736:
4733:
4710:
4704:
4699:
4670:
4658:
4635:
4629:
4626:
4613:
4597:{\textstyle x}
4593:
4573:{\textstyle x}
4569:
4553:{\textstyle x}
4549:
4533:{\textstyle k}
4529:
4509:
4506:
4503:
4500:
4495:
4491:
4487:
4484:
4481:
4478:
4475:
4472:
4467:
4448:{\textstyle f}
4444:
4428:{\textstyle p}
4424:
4404:
4401:
4398:
4395:
4392:
4389:
4380:and a density
4369:
4366:
4363:
4358:
4354:
4340:
4337:
4312:
4309:
4294:
4290:
4267:
4263:
4242:
4239:
4236:
4233:
4228:
4224:
4220:
4217:
4214:
4209:
4203:
4200:
4197:
4193:
4190:
4187:
4181:
4178:
4173:
4169:
4165:
4162:
4159:
4154:
4149:
4144:
4138:
4135:
4132:
4128:
4125:
4122:
4093:
4090:
4087:
4084:
4079:
4075:
4071:
4068:
4065:
4060:
4054:
4051:
4048:
4044:
4041:
4038:
4032:
4028:
4024:
4021:
4017:
4014:
4011:
4008:
4005:
4002:
3997:
3994:
3989:
3985:
3977:
3973:
3968:
3962:
3959:
3950:
3946:
3942:
3939:
3935:
3930:
3926:
3921:
3915:
3912:
3909:
3905:
3902:
3899:
3893:
3889:
3886:
3885:
3881:
3877:
3874:
3871:
3868:
3863:
3859:
3855:
3850:
3846:
3842:
3839:
3836:
3833:
3830:
3827:
3822:
3813:
3809:
3805:
3802:
3798:
3793:
3789:
3784:
3778:
3775:
3772:
3768:
3765:
3762:
3756:
3753:
3748:
3744:
3740:
3737:
3734:
3729:
3724:
3719:
3713:
3710:
3707:
3703:
3700:
3697:
3691:
3689:
3667:
3664:
3661:
3658:
3653:
3649:
3645:
3642:
3639:
3636:
3633:
3629:
3626:
3623:
3620:
3617:
3614:
3609:
3606:
3601:
3597:
3589:
3585:
3580:
3574:
3571:
3562:
3558:
3554:
3551:
3547:
3537:provides that
3520:
3517:
3514:
3510:
3507:
3504:
3501:
3498:
3495:
3490:
3487:
3482:
3478:
3470:
3466:
3461:
3455:
3452:
3447:
3442:
3436:
3433:
3430:
3426:
3423:
3420:
3414:
3411:
3408:
3405:
3402:
3397:
3393:
3389:
3384:
3380:
3376:
3373:
3370:
3367:
3364:
3361:
3356:
3351:
3346:
3340:
3337:
3334:
3330:
3327:
3324:
3299:
3296:
3293:
3290:
3287:
3284:
3262:
3259:
3256:
3252:
3249:
3246:
3243:
3240:
3237:
3232:
3229:
3224:
3220:
3212:
3208:
3203:
3197:
3194:
3189:
3184:
3178:
3175:
3172:
3168:
3165:
3162:
3156:
3153:
3150:
3147:
3144:
3141:
3136:
3132:
3128:
3125:
3122:
3117:
3113:
3109:
3106:
3101:
3098:
3093:
3088:
3082:
3079:
3076:
3072:
3069:
3066:
3060:
3057:
3054:
3051:
3048:
3043:
3039:
3035:
3030:
3026:
3022:
3019:
3016:
3013:
3010:
3007:
3002:
2995:
2992:
2987:
2982:
2976:
2973:
2970:
2966:
2963:
2960:
2941:{\textstyle h}
2937:
2917:
2914:
2911:
2908:
2905:
2900:
2896:
2892:
2887:
2883:
2879:
2876:
2873:
2870:
2867:
2864:
2859:
2852:
2849:
2844:
2839:
2833:
2830:
2827:
2823:
2820:
2817:
2811:
2808:
2805:
2802:
2799:
2794:
2790:
2786:
2781:
2777:
2773:
2770:
2767:
2764:
2761:
2758:
2753:
2748:
2743:
2737:
2734:
2731:
2727:
2724:
2721:
2698:
2695:
2692:
2689:
2684:
2680:
2676:
2671:
2667:
2663:
2660:
2657:
2654:
2651:
2648:
2643:
2621:
2618:
2615:
2595:
2592:
2589:
2584:
2580:
2576:
2571:
2567:
2563:
2541:
2537:
2520:
2517:
2504:
2501:
2498:
2478:
2458:
2436:
2414:
2411:
2408:
2385:
2361:
2339:
2336:
2333:
2330:
2325:
2321:
2317:
2314:
2311:
2308:
2305:
2302:
2297:
2273:
2257:{\textstyle x}
2253:
2237:{\textstyle f}
2233:
2206:{\textstyle X}
2202:
2190:
2187:
2172:
2144:
2121:
2118:
2115:
2107:
2104:
2101:
2092:
2088:
2083:
2057:
2054:
2049:
2045:
2041:
2038:
2035:
2032:
2023:
2019:
2011:
2008:
1986:
1983:
1974:
1950:
1942:
1939:
1919:
1916:
1913:
1904:
1900:
1897:
1886:Bayes' theorem
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1843:
1839:
1836:
1814:
1811:
1808:
1800:
1797:
1794:
1785:
1781:
1776:
1752:
1749:
1740:
1717:
1714:
1709:
1705:
1701:
1698:
1695:
1692:
1683:
1679:
1671:
1668:
1639:
1636:
1627:
1596:
1569:
1544:
1535:
1531:
1528:
1525:
1520:
1511:
1480:
1471:
1454:
1451:
1442:{\textstyle x}
1438:
1418:
1398:
1395:
1392:
1389:
1386:
1383:
1363:
1360:
1357:
1354:
1351:
1346:
1324:
1304:{\textstyle x}
1300:
1280:
1264:{\textstyle x}
1260:
1199:
1183:{\textstyle X}
1179:
1163:{\textstyle x}
1159:
1143:{\textstyle X}
1139:
1123:{\textstyle x}
1119:
1092:
1070:
1067:
1064:
1061:
1058:
1055:
1050:
1046:
1042:
1039:
1036:
1033:
1028:
1024:
1020:
1017:
1014:
1011:
1008:
1005:
1000:
976:
960:{\textstyle p}
956:
940:be a discrete
933:{\textstyle X}
929:
917:
914:
897:
894:
891:
871:
840:
820:
817:
814:
811:
808:
805:
785:
782:
779:
776:
773:
770:
750:
747:
744:
741:
738:
735:
715:{\textstyle x}
711:
691:
671:
655:{\textstyle x}
651:
631:
628:
625:
622:
619:
616:
594:
591:
588:
585:
582:
579:
574:
563:often written
552:
549:
546:
543:
540:
537:
534:
531:
528:
512:{\textstyle X}
508:
492:{\textstyle x}
488:
466:
463:
460:
457:
454:
451:
448:
445:
442:
413:
401:
398:
371:Hessian matrix
363:point estimate
318:
317:
315:
314:
307:
300:
292:
289:
288:
287:
286:
271:
270:
269:
268:
263:
258:
250:
249:
245:
244:
243:
242:
237:
229:
228:
224:
223:
222:
221:
216:
211:
203:
202:
198:
197:
196:
195:
190:
185:
180:
175:
167:
166:
162:
161:
160:
159:
154:
149:
144:
136:
135:
134:Model building
131:
130:
129:
128:
123:
118:
113:
108:
103:
98:
93:
91:Bayes' theorem
88:
83:
75:
74:
70:
69:
51:
50:
42:
41:
35:
34:
26:
18:Log-likelihood
9:
6:
4:
3:
2:
15519:
15508:
15505:
15503:
15500:
15499:
15497:
15487:
15482:
15477:
15476:
15473:
15460:
15459:
15450:
15448:
15447:
15438:
15436:
15435:
15430:
15424:
15422:
15421:
15412:
15411:
15408:
15394:
15391:
15389:
15388:Geostatistics
15386:
15384:
15381:
15379:
15376:
15374:
15371:
15370:
15368:
15366:
15362:
15356:
15355:Psychometrics
15353:
15351:
15348:
15346:
15343:
15341:
15338:
15336:
15333:
15331:
15328:
15326:
15323:
15321:
15318:
15316:
15313:
15311:
15308:
15307:
15305:
15303:
15299:
15293:
15290:
15288:
15285:
15283:
15279:
15276:
15274:
15271:
15269:
15266:
15264:
15261:
15260:
15258:
15256:
15252:
15246:
15243:
15241:
15238:
15236:
15232:
15229:
15227:
15224:
15223:
15221:
15219:
15218:Biostatistics
15215:
15211:
15207:
15202:
15198:
15180:
15179:Log-rank test
15177:
15176:
15174:
15170:
15164:
15161:
15160:
15158:
15156:
15152:
15146:
15143:
15141:
15138:
15136:
15133:
15131:
15128:
15127:
15125:
15123:
15119:
15116:
15114:
15110:
15100:
15097:
15095:
15092:
15090:
15087:
15085:
15082:
15080:
15077:
15076:
15074:
15072:
15068:
15062:
15059:
15057:
15054:
15052:
15050:(BoxâJenkins)
15046:
15044:
15041:
15039:
15036:
15032:
15029:
15028:
15027:
15024:
15023:
15021:
15019:
15015:
15009:
15006:
15004:
15003:DurbinâWatson
15001:
14999:
14993:
14991:
14988:
14986:
14985:DickeyâFuller
14983:
14982:
14980:
14976:
14970:
14967:
14965:
14962:
14960:
14959:Cointegration
14957:
14955:
14952:
14950:
14947:
14945:
14942:
14940:
14937:
14935:
14934:Decomposition
14932:
14931:
14929:
14925:
14922:
14920:
14916:
14906:
14903:
14902:
14901:
14898:
14897:
14896:
14893:
14889:
14886:
14885:
14884:
14881:
14879:
14876:
14874:
14871:
14869:
14866:
14864:
14861:
14859:
14856:
14854:
14851:
14849:
14846:
14845:
14843:
14841:
14837:
14831:
14828:
14826:
14823:
14821:
14818:
14816:
14813:
14811:
14808:
14806:
14805:Cohen's kappa
14803:
14802:
14800:
14798:
14794:
14790:
14786:
14782:
14778:
14774:
14769:
14765:
14751:
14748:
14746:
14743:
14741:
14738:
14736:
14733:
14732:
14730:
14728:
14724:
14718:
14714:
14710:
14704:
14702:
14699:
14698:
14696:
14694:
14690:
14684:
14681:
14679:
14676:
14674:
14671:
14669:
14666:
14664:
14661:
14659:
14658:Nonparametric
14656:
14654:
14651:
14650:
14648:
14644:
14638:
14635:
14633:
14630:
14628:
14625:
14623:
14620:
14619:
14617:
14615:
14611:
14605:
14602:
14600:
14597:
14595:
14592:
14590:
14587:
14585:
14582:
14581:
14579:
14577:
14573:
14567:
14564:
14562:
14559:
14557:
14554:
14552:
14549:
14548:
14546:
14544:
14540:
14536:
14529:
14526:
14524:
14521:
14520:
14516:
14512:
14496:
14493:
14492:
14491:
14488:
14486:
14483:
14481:
14478:
14474:
14471:
14469:
14466:
14465:
14464:
14461:
14460:
14458:
14456:
14452:
14442:
14439:
14435:
14429:
14427:
14421:
14419:
14413:
14412:
14411:
14408:
14407:Nonparametric
14405:
14403:
14397:
14393:
14390:
14389:
14388:
14382:
14378:
14377:Sample median
14375:
14374:
14373:
14370:
14369:
14367:
14365:
14361:
14353:
14350:
14348:
14345:
14343:
14340:
14339:
14338:
14335:
14333:
14330:
14328:
14322:
14320:
14317:
14315:
14312:
14310:
14307:
14305:
14302:
14300:
14298:
14294:
14292:
14289:
14288:
14286:
14284:
14280:
14274:
14272:
14268:
14266:
14264:
14259:
14257:
14252:
14248:
14247:
14244:
14241:
14239:
14235:
14225:
14222:
14220:
14217:
14215:
14212:
14211:
14209:
14207:
14203:
14197:
14194:
14190:
14187:
14186:
14185:
14182:
14178:
14175:
14174:
14173:
14170:
14168:
14165:
14164:
14162:
14160:
14156:
14148:
14145:
14143:
14140:
14139:
14138:
14135:
14133:
14130:
14128:
14125:
14123:
14120:
14118:
14115:
14113:
14110:
14109:
14107:
14105:
14101:
14095:
14092:
14088:
14085:
14081:
14078:
14076:
14073:
14072:
14071:
14068:
14067:
14066:
14063:
14059:
14056:
14054:
14051:
14049:
14046:
14044:
14041:
14040:
14039:
14036:
14035:
14033:
14031:
14027:
14024:
14022:
14018:
14012:
14009:
14007:
14004:
14000:
13997:
13996:
13995:
13992:
13990:
13987:
13983:
13982:loss function
13980:
13979:
13978:
13975:
13971:
13968:
13966:
13963:
13961:
13958:
13957:
13956:
13953:
13951:
13948:
13946:
13943:
13939:
13936:
13934:
13931:
13929:
13923:
13920:
13919:
13918:
13915:
13911:
13908:
13906:
13903:
13901:
13898:
13897:
13896:
13893:
13889:
13886:
13884:
13881:
13880:
13879:
13876:
13872:
13869:
13868:
13867:
13864:
13860:
13857:
13856:
13855:
13852:
13850:
13847:
13845:
13842:
13840:
13837:
13836:
13834:
13832:
13828:
13824:
13820:
13815:
13811:
13797:
13794:
13792:
13789:
13787:
13784:
13782:
13779:
13778:
13776:
13774:
13770:
13764:
13761:
13759:
13756:
13754:
13751:
13750:
13748:
13744:
13738:
13735:
13733:
13730:
13728:
13725:
13723:
13720:
13718:
13715:
13713:
13710:
13708:
13705:
13704:
13702:
13700:
13696:
13690:
13687:
13685:
13684:Questionnaire
13682:
13680:
13677:
13673:
13670:
13668:
13665:
13664:
13663:
13660:
13659:
13657:
13655:
13651:
13645:
13642:
13640:
13637:
13635:
13632:
13630:
13627:
13625:
13622:
13620:
13617:
13615:
13612:
13610:
13607:
13606:
13604:
13602:
13598:
13594:
13590:
13585:
13581:
13567:
13564:
13562:
13559:
13557:
13554:
13552:
13549:
13547:
13544:
13542:
13539:
13537:
13534:
13532:
13529:
13527:
13524:
13522:
13519:
13517:
13514:
13512:
13511:Control chart
13509:
13507:
13504:
13502:
13499:
13497:
13494:
13493:
13491:
13489:
13485:
13479:
13476:
13472:
13469:
13467:
13464:
13463:
13462:
13459:
13457:
13454:
13452:
13449:
13448:
13446:
13444:
13440:
13434:
13431:
13429:
13426:
13424:
13421:
13420:
13418:
13414:
13408:
13405:
13404:
13402:
13400:
13396:
13384:
13381:
13379:
13376:
13374:
13371:
13370:
13369:
13366:
13364:
13361:
13360:
13358:
13356:
13352:
13346:
13343:
13341:
13338:
13336:
13333:
13331:
13328:
13326:
13323:
13321:
13318:
13316:
13313:
13312:
13310:
13308:
13304:
13298:
13295:
13293:
13290:
13286:
13283:
13281:
13278:
13276:
13273:
13271:
13268:
13266:
13263:
13261:
13258:
13256:
13253:
13251:
13248:
13246:
13243:
13241:
13238:
13237:
13236:
13233:
13232:
13230:
13228:
13224:
13221:
13219:
13215:
13211:
13207:
13202:
13198:
13192:
13189:
13187:
13184:
13183:
13180:
13176:
13169:
13164:
13162:
13157:
13155:
13150:
13149:
13146:
13138:
13134:
13130:
13128:
13125:
13124:
13114:
13108:
13104:
13100:
13096:
13092:
13088:
13084:
13082:0-412-04411-0
13078:
13073:
13072:
13065:
13061:
13055:
13051:
13046:
13042:
13040:0-19-852359-9
13036:
13032:
13028:
13023:
13018:
13013:
13008:
13003:
12999:
12995:
12991:
12986:
12982:
12980:0-521-36697-6
12976:
12972:
12968:
12964:
12960:
12956:
12954:0-8018-4443-6
12950:
12946:
12942:
12938:
12934:
12930:
12924:
12920:
12916:
12912:
12907:
12903:
12901:0-412-60650-X
12897:
12893:
12888:
12887:
12874:
12870:
12863:
12855:
12851:
12844:
12836:
12832:
12828:
12822:
12815:
12813:9781118341544
12809:
12805:
12801:
12797:
12790:
12783:
12777:
12775:
12773:
12771:
12763:
12757:
12755:
12753:
12751:
12743:
12737:
12735:
12733:
12731:
12729:
12721:
12718:H. Jeffreys:
12715:
12713:
12711:
12709:
12701:
12695:
12693:
12691:
12689:
12680:
12676:
12669:
12661:
12657:
12650:
12641:
12636:
12632:
12628:
12627:
12619:
12611:
12607:
12603:
12599:
12595:
12591:
12584:
12576:
12572:
12565:
12557:
12553:
12549:
12545:
12540:
12535:
12530:
12525:
12521:
12517:
12513:
12509:
12505:
12501:
12495:
12487:
12483:
12479:
12473:
12465:
12461:
12456:
12451:
12447:
12443:
12442:
12437:
12433:
12427:
12420:
12419:
12412:
12404:
12400:
12393:
12385:
12381:
12377:
12373:
12372:
12364:
12356:
12352:
12348:
12344:
12337:
12329:
12325:
12321:
12317:
12310:
12302:
12298:
12294:
12290:
12289:
12281:
12273:
12271:0-8018-4443-6
12267:
12263:
12259:
12255:
12249:
12247:
12238:
12237:
12232:
12225:
12217:
12215:0-471-82668-5
12211:
12207:
12206:
12198:
12190:
12186:
12182:
12178:
12174:
12170:
12169:
12164:
12158:
12150:
12146:
12142:
12138:
12131:
12123:
12119:
12115:
12111:
12107:
12103:
12102:
12094:
12086:
12084:0-387-90777-7
12080:
12076:
12072:
12066:
12058:
12052:
12048:
12047:
12039:
12031:
12029:0-86094-190-6
12025:
12021:
12017:
12016:
12008:
12000:
11994:
11990:
11986:
11979:
11971:
11965:
11961:
11957:
11950:
11942:
11936:
11932:
11927:
11926:
11920:
11916:
11910:
11895:
11891:
11884:
11876:
11872:
11865:
11856:
11852:
11851:
11843:
11841:
11832:, p. 267
11831:
11827:
11820:
11818:
11816:
11806:
11799:
11792:
11788:
11782:
11775:
11769:
11767:
11765:
11756:
11754:9780412606502
11750:
11746:
11742:
11741:
11733:
11724:
11720:
11714:
11712:
11710:
11708:
11699:
11695:
11691:
11687:
11686:
11678:
11670:
11668:0-444-88376-2
11664:
11660:
11653:
11645:
11641:
11637:
11633:
11626:
11618:
11614:
11610:
11606:
11599:
11591:
11589:0-471-09077-8
11585:
11581:
11574:
11566:
11562:
11558:
11554:
11550:
11546:
11545:
11537:
11529:
11525:
11521:
11517:
11513:
11509:
11502:
11494:
11490:
11485:
11480:
11476:
11472:
11471:
11466:
11459:
11451:
11449:0-521-40551-3
11445:
11441:
11440:
11435:
11429:
11421:
11414:
11406:
11402:
11398:
11392:
11384:
11382:0-471-98165-6
11378:
11374:
11367:
11359:
11357:0-387-98502-6
11353:
11349:
11342:
11334:
11328:
11324:
11317:
11309:
11307:0-534-24312-6
11303:
11299:
11292:
11288:
11275:
11269:
11265:
11254:
11251:
11249:
11246:
11244:
11241:
11239:
11236:
11234:
11231:
11229:
11226:
11224:
11221:
11219:
11216:
11214:
11211:
11209:
11206:
11204:
11201:
11200:
11193:
11191:
11187:
11176:
11167:
11163:
11160:This section
11158:
11155:
11151:
11150:
11142:
11138:
11135:
11131:
11124:
11117:
11113:
11107:
11105:
11104:Ď distributed
11100:
11096:
11092:
11088:
11087:
11081:
11079:
11075:
11068:
11061:
11057:
11046:
11043:
11035:
11025:
11021:
11015:
11014:
11008:
11003:
10994:
10993:
10985:
10983:
10970:
10950:
10928:
10908:
10900:
10896:
10892:
10888:
10883:
10879:
10875:
10871:
10867:
10851:
10842:
10838:
10834:
10831:
10827:
10826:
10818:
10816:
10812:
10811:likelihoodism
10808:
10804:
10800:
10790:
10788:
10787:phylogenetics
10784:
10780:
10776:
10774:
10768:
10763:
10759:
10754:
10752:
10748:
10747:Ronald Fisher
10744:
10740:
10734:
10730:
10715:
10713:
10694:
10690:
10684:
10679:
10676:
10673:
10669:
10663:
10660:
10655:
10646:
10619:
10616:
10604:
10591:
10581:
10576:
10571:
10565:
10562:
10551:
10537:
10528:
10511:
10506:
10502:
10496:
10491:
10488:
10485:
10481:
10477:
10472:
10468:
10465:
10459:
10453:
10440:
10436:
10432:
10429:
10426:
10423:
10410:
10407:
10398:
10395:
10392:
10386:
10373:
10369:
10365:
10362:
10359:
10356:
10343:
10340:
10327:
10317:
10304:
10300:
10296:
10293:
10290:
10285:
10281:
10277:
10274:
10271:
10268:
10255:
10252:
10233:
10217:
10213:
10209:
10206:
10203:
10198:
10194:
10184:
10171:
10168:
10165:
10160:
10157:
10152:
10146:
10135:
10132:
10129:
10126:
10123:
10110:
10107:
10093:
10079:
10071:
10066:
10052:
10049:
10046:
10043:
10040:
10037:
10034:
10028:
10025:
10022:
10016:
10010:
10001:
9998:
9995:
9992:
9989:
9986:
9983:
9980:
9974:
9971:
9968:
9965:
9962:
9949:
9946:
9938:
9924:
9904:
9895:
9882:
9877:
9874:
9871:
9867:
9861:
9858:
9855:
9851:
9841:
9829:
9825:
9819:
9813:
9810:
9807:
9804:
9801:
9783:
9769:
9749:
9741:
9731:
9728:
9722:
9692:
9682:
9661:
9644:
9613:
9600:
9586:
9583:
9574:
9563:
9552:
9546:
9543:
9532:
9524:
9522:
9503:
9497:
9455:
9442:
9436:
9430:
9427:
9424:
9421:
9407:
9404:
9395:
9384:
9362:
9356:
9353:
9342:
9334:
9331:
9318:
9297:
9294:
9285:
9274:
9245:
9242:
9236:
9230:
9227:
9216:
9213:
9207:
9199:
9197:
9196:inner product
9178:
9175:
9172:
9160:
9158:
9154:
9150:
9144:
9134:
9132:
9128:
9106:
9094:
9092:
9074:
9070:
9060:
9054:
9044:
9032:
9031:almost surely
9011:
9003:
8993:
8981:
8977:
8955:
8950:
8940:
8933:
8912:
8879:
8876:
8871:
8867:
8863:
8858:
8848:
8814:
8810:
8792:
8789:
8784:
8780:
8771:
8747:
8744:-dimensional
8725:
8690:
8680:
8675:
8672:
8667:
8663:
8655:
8612:
8606:
8598:
8594:
8585:
8580:
8578:
8574:
8570:
8566:
8547:
8539:
8535:
8529:
8521:
8515:
8507:
8503:
8494:
8490:
8486:
8476:
8474:
8470:
8466:
8462:
8457:
8455:
8451:
8446:
8444:
8440:
8436:
8432:
8428:
8427:support curve
8424:
8414:
8411:
8398:
8392:
8386:
8383:
8377:
8371:
8368:
8362:
8349:
8346:
8343:
8337:
8324:
8321:
8318:
8309:
8291:
8275:
8272:
8263:
8261:
8257:
8253:
8249:
8244:
8239:
8237:
8233:
8229:
8225:
8221:
8218:ânotably the
8217:
8213:
8209:
8180:
8161:
8150:
8145:
8140:
8130:
8128:
8124:
8123:uniform prior
8119:
8117:
8113:
8108:
8094:
8086:
8082:
8078:
8075:
8066:
8058:
8054:
8050:
8047:
8038:
8030:
8026:
8022:
8017:
8013:
8009:
8006:
7992:
7989:
7979:
7977:
7967:
7965:
7961:
7954:
7944:
7942:
7938:
7933:
7931:
7921:
7919:
7915:
7911:
7893:
7889:
7880:
7877:
7859:
7855:
7845:
7843:
7825:
7810:
7785:
7773:
7770:
7765:
7759:
7740:
7729:
7722:
7712:
7707:
7674:
7668:
7658:
7649:
7636:
7624:
7621:
7616:
7610:
7599:
7593:
7583:
7574:
7561:
7550:
7545:
7540:
7530:
7504:
7500:
7478:
7472:
7468:
7462:
7452:
7443:
7430:
7418:
7415:
7410:
7404:
7385:
7374:
7369:
7361:
7357:
7348:
7344:
7321:
7317:
7295:
7289:
7279:
7274:
7263:
7259:
7247:
7246:design matrix
7230:
7224:
7220:
7216:
7211:
7207:
7202:
7198:
7195:
7187:
7171:
7168:
7165:
7157:
7144:
7139:
7137:
7120:
7115:
7110:
7105:
7099:
7089:
7082:
7077:
7067:
7042:
7036:
7031:
7026:
7021:
7016:
7010:
7006:
7002:
6980:
6964:
6962:
6958:
6948:
6942:
6938:
6925:
6923:
6919:
6915:
6911:
6907:
6902:
6900:
6896:
6883:
6870:
6866:
6860:
6857:
6852:
6846:
6840:
6837:
6834:
6830:
6821:
6815:
6804:
6794:
6771:
6738:
6729:
6726:
6717:
6696:
6693:
6690:
6674:
6668:
6662:
6650:
6628:
6595:
6580:
6574:
6564:
6562:
6558:
6554:
6549:
6546:
6533:
6527:
6524:
6519:
6515:
6511:
6506:
6502:
6492:
6484:
6480:
6476:
6471:
6467:
6460:
6457:
6451:
6448:
6443:
6439:
6435:
6430:
6426:
6419:
6411:
6393:
6367:
6363:
6336:
6332:
6321:
6317:
6313:
6309:
6305:
6300:
6298:
6294:
6290:
6286:
6282:
6278:
6274:
6270:
6265:
6263:
6262:
6257:
6252:
6239:
6230:
6227:
6222:
6218:
6199:
6196:
6191:
6187:
6170:
6164:
6161:
6156:
6152:
6148:
6143:
6139:
6124:
6117:
6113:
6102:
6092:
6090:
6086:
6081:
6077:
6075:
6057:
6050:
6038:
6028:
6012:
6001:
5990:
5986:
5977:
5974:
5971:
5954:
5950:
5941:
5938:
5935:
5916:
5912:
5908:
5902:
5886:
5882:
5866:
5859:
5856:
5853:
5850:
5839:
5831:
5828:
5825:
5821:
5807:
5803:
5794:is such that
5781:
5758:
5750:
5747:
5744:
5740:
5736:
5732:
5724:
5720:
5710:
5706:
5696:
5692:
5683:
5678:
5667:
5662:
5655:
5647:
5644:
5640:
5636:
5632:
5624:
5620:
5610:
5606:
5597:
5592:
5581:
5576:
5569:
5561:
5557:
5553:
5549:
5541:
5537:
5528:
5519:
5494:
5491:
5470:
5462:
5445:
5442:
5439:
5436:
5433:
5430:
5427:
5424:
5421:
5418:
5415:
5412:
5409:
5382:
5378:
5368:
5364:
5354:
5350:
5341:
5338:
5335:
5330:
5318:
5309:
5305:
5296:
5292:
5283:
5280:
5277:
5272:
5260:
5251:
5247:
5238:
5235:
5232:
5207:
5200:
5197:
5176:
5169:
5164:
5159:
5157:
5153:
5115:
5111:
5108:
5102:
5096:
5082:
5057:
5041:
5014:
5008:
5005:
5002:
4997:
4986:
4982:
4972:
4961:
4956:
4953:
4925:
4922:
4913:
4886:
4882:
4871:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4842:
4831:
4827:
4817:
4813:
4803:
4798:
4785:
4780:
4774:
4759:
4740:
4731:
4708:
4702:
4687:
4684:
4655:
4653:
4648:
4644:
4640:
4633:
4625:
4611:
4591:
4581:
4567:
4547:
4527:
4507:
4501:
4493:
4489:
4485:
4479:
4476:
4473:
4442:
4422:
4399:
4396:
4393:
4387:
4364:
4356:
4352:
4336:
4334:
4329:
4326:
4322:
4318:
4308:
4292:
4288:
4265:
4261:
4240:
4234:
4231:
4226:
4222:
4215:
4212:
4207:
4179:
4171:
4167:
4163:
4160:
4147:
4142:
4108:
4091:
4085:
4082:
4077:
4073:
4066:
4063:
4058:
4030:
4026:
4022:
4019:
4012:
4009:
4006:
4000:
3995:
3992:
3987:
3983:
3975:
3971:
3966:
3960:
3957:
3948:
3944:
3937:
3928:
3924:
3919:
3887:
3879:
3869:
3866:
3861:
3857:
3853:
3848:
3844:
3837:
3834:
3831:
3828:
3811:
3807:
3800:
3791:
3787:
3782:
3754:
3746:
3742:
3738:
3735:
3722:
3717:
3678:
3665:
3659:
3656:
3651:
3647:
3640:
3637:
3634:
3631:
3624:
3621:
3618:
3612:
3607:
3604:
3599:
3595:
3587:
3583:
3578:
3572:
3569:
3560:
3556:
3549:
3536:
3531:
3518:
3515:
3512:
3505:
3502:
3499:
3493:
3488:
3485:
3480:
3476:
3468:
3464:
3459:
3453:
3450:
3445:
3440:
3412:
3403:
3400:
3395:
3391:
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2599:{\textstyle }
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349:
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341:
337:
336:observed data
333:
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119:
117:
114:
112:
109:
107:
106:Cox's theorem
104:
102:
99:
97:
94:
92:
89:
87:
84:
82:
79:
78:
77:
76:
72:
71:
68:
64:
60:
56:
53:
52:
48:
44:
43:
40:
37:
36:
32:
31:
19:
15456:
15444:
15425:
15418:
15330:Econometrics
15280: /
15263:Chemometrics
15240:Epidemiology
15233: /
15206:Applications
15048:ARIMA model
14995:Q-statistic
14944:Stationarity
14840:Multivariate
14783: /
14779: /
14777:Multivariate
14775: /
14715: /
14711: /
14485:Bayes factor
14384:Signed rank
14296:
14270:
14262:
14250:
13945:Completeness
13921:
13781:Cohort study
13679:Opinion poll
13614:Missing data
13601:Study design
13556:Scatter plot
13478:Scatter plot
13471:Spearman's Ď
13433:Grouped data
13136:
13098:
13070:
13049:
13030:
13027:"Likelihood"
13017:10419/258120
12997:
12993:
12970:
12940:
12910:
12891:
12868:
12862:
12849:
12843:
12834:
12821:
12795:
12789:
12781:
12761:
12741:
12719:
12699:
12698:I. J. Good:
12674:
12668:
12655:
12649:
12630:
12624:
12618:
12593:
12589:
12583:
12570:
12564:
12511:
12507:
12500:Fisher, R.A.
12494:
12485:
12481:
12478:Fisher, R.A.
12472:
12445:
12439:
12426:
12416:
12411:
12398:
12392:
12375:
12369:
12363:
12346:
12342:
12336:
12319:
12315:
12309:
12292:
12286:
12280:
12257:
12234:
12224:
12204:
12197:
12172:
12166:
12157:
12140:
12139:. Series A.
12136:
12130:
12108:(1): 87â94.
12105:
12099:
12093:
12074:
12065:
12045:
12038:
12014:
12007:
11988:
11978:
11959:
11949:
11924:
11909:
11898:. Retrieved
11883:
11870:
11864:
11857:(2): 256â262
11854:
11848:
11825:
11804:
11798:
11786:
11781:
11773:
11739:
11732:
11722:
11689:
11683:
11677:
11658:
11652:
11635:
11631:
11625:
11608:
11604:
11598:
11579:
11573:
11548:
11542:
11536:
11511:
11508:Optimization
11507:
11501:
11474:
11468:
11458:
11438:
11428:
11419:
11413:
11400:
11391:
11372:
11366:
11347:
11341:
11322:
11316:
11297:
11291:
11268:
11203:Bayes factor
11183:
11170:
11166:adding to it
11161:
11139:
11122:
11115:
11108:
11094:
11085:
11084:
11082:
11073:
11066:
11059:
11053:
11038:
11029:
11010:
10942:
10886:
10863:
10845:
10841:adding to it
10836:
10796:
10782:
10777:
10770:
10765:
10761:
10756:
10736:
10605:
10552:
10529:
10234:
10185:
10094:
10067:
9939:
9896:
9784:
9737:
9726:
9720:
9614:
9525:
9456:
9335:
9332:
9200:
9161:
9146:
9095:
8809:well-defined
8581:
8577:product rule
8495:and written
8482:
8458:
8447:
8434:
8426:
8420:
8412:
8264:
8251:
8240:
8236:maximization
8178:
8148:
8143:
8142:
8120:
8109:
7985:
7973:
7960:mixed models
7956:
7934:
7927:
7909:
7846:
7140:
6970:
6954:
6926:
6903:
6898:
6884:
6822:
6810:
6802:
6800:
6648:
6576:
6550:
6547:
6409:
6319:
6308:Bayes factor
6301:
6266:
6259:
6253:
6122:
6120:
6082:
6078:
5160:
5152:Morse theory
4656:
4631:
4584:observation
4582:
4342:
4330:
4314:
4109:
3679:
3532:
3312:
3274:
2522:
2396:, given the
2373:
2351:
2286:
2192:
2133:
2072:
2069:
1999:
1963:
1826:
1765:
1729:
1659:
1652:
1558:
1312:
1246:
1242:
1238:
1234:
1227:
1223:
1219:
1215:
1107:, given the
1104:
1082:
989:
919:
859:
857:
606:
478:
433:
403:
385:
379:
352:
327:
323:
321:
256:Bayes factor
58:
15486:Mathematics
15458:WikiProject
15373:Cartography
15335:Jurimetrics
15287:Reliability
15018:Time domain
14997:(LjungâBox)
14919:Time-series
14797:Categorical
14781:Time-series
14773:Categorical
14708:(Bernoulli)
14543:Correlation
14523:Correlation
14319:JarqueâBera
14291:Chi-squared
14053:M-estimator
14006:Asymptotics
13950:Sufficiency
13717:Interaction
13629:Replication
13609:Effect size
13566:Violin plot
13546:Radar chart
13526:Forest plot
13516:Correlogram
13466:Kendall's Ď
11024:introducing
10891:Bayes' Rule
10887:probability
10807:Bayesianism
10803:frequentism
10712:sample mean
9125:, known as
7988:independent
7186:partitioned
6312:Bayes' rule
6291:at a given
5163:consistency
4110:Therefore,
394:Bayes' rule
15502:Likelihood
15496:Categories
15325:Demography
15043:ARMA model
14848:Regression
14425:(Friedman)
14386:(Wilcoxon)
14324:Normality
14314:Lilliefors
14261:Student's
14137:Resampling
14011:Robustness
13999:divergence
13989:Efficiency
13927:(monotone)
13922:Likelihood
13839:Population
13672:Stratified
13624:Population
13443:Dependence
13399:Count data
13330:Percentile
13307:Dispersion
13240:Arithmetic
13175:Statistics
12963:King, Gary
12941:Likelihood
12827:Akaike, H.
12633:(3): 161.
12548:48.1280.02
12539:2440/15172
12371:Biometrika
12258:Likelihood
12168:Biometrika
12163:Cox, D. R.
11900:2017-10-01
11807:, Springer
11725:, Springer
11544:Biometrika
11283:References
11184:Under the
11173:March 2019
11032:April 2019
11007:references
10848:March 2019
10727:See also:
8969:such that
8573:derivative
8431:univariate
8222:âare only
8137:See also:
6571:See also:
6099:See also:
5168:almost all
4914:for every
4688:subset of
4643:continuous
4311:In general
3533:The first
2425:). Again,
400:Definition
328:likelihood
201:Estimators
73:Background
59:Likelihood
14706:Logistic
14473:posterior
14399:Rank sum
14147:Jackknife
14142:Bootstrap
13960:Bootstrap
13895:Parameter
13844:Statistic
13639:Statistic
13551:Run chart
13536:Pie chart
13531:Histogram
13521:Fan chart
13496:Bar chart
13378:L-moments
13265:Geometric
13000:(2): 31.
12939:(1992) .
12854:D. Reidel
12256:(1992) .
11793:(§4.1.2).
11758:(§1.4.2).
11132:given by
11056:statistic
10815:AIC-based
10670:∑
10650:¯
10620:^
10617:β
10585:¯
10577:α
10566:^
10563:β
10538:β
10482:∑
10478:−
10473:β
10469:α
10454:β
10451:∂
10433:∣
10430:β
10424:α
10411:
10405:∂
10396:⋯
10387:β
10384:∂
10366:∣
10363:β
10357:α
10344:
10338:∂
10318:β
10315:∂
10294:…
10278:∣
10275:β
10269:α
10256:
10250:∂
10207:…
10166:−
10161:β
10158:α
10147:β
10144:∂
10133:∣
10130:β
10124:α
10111:
10105:∂
10080:β
10047:β
10044:−
10038:
10026:−
10023:α
10011:α
10005:Γ
10002:
9996:−
9993:β
9990:
9984:α
9972:∣
9969:β
9963:α
9950:
9905:β
9875:β
9872:−
9859:−
9856:α
9842:α
9836:Γ
9830:α
9826:β
9811:∣
9808:β
9802:α
9770:β
9750:α
9700:η
9626:η
9594:η
9584:−
9581:⟩
9560:η
9556:⟨
9544:∣
9540:η
9533:ℓ
9474:θ
9466:η
9428:
9415:θ
9405:−
9402:⟩
9378:θ
9370:η
9366:⟨
9354:∣
9350:θ
9343:ℓ
9305:θ
9295:−
9292:⟩
9268:θ
9260:η
9256:⟨
9246:
9221:θ
9217:∣
9182:⟩
9179:−
9173:−
9170:⟨
9131:precision
9110:^
9107:θ
9071:θ
9048:^
9045:θ
8997:^
8994:θ
8944:^
8941:θ
8913:θ
8877:−
8852:^
8849:θ
8790:−
8756:Θ
8699:Θ
8696:→
8673:−
8607:θ
8548:θ
8536:ℓ
8530:θ
8526:∇
8522:≡
8516:θ
8437:over the
8387:ℓ
8384:−
8372:ℓ
8350:
8344:−
8325:
8276:
8260:surprisal
8228:concavity
8162:ℓ
8079:∣
8070:Λ
8067:⋅
8051:∣
8042:Λ
8023:∧
8010:∣
8001:Λ
7890:β
7876:isometric
7856:β
7771:−
7659:−
7622:−
7584:−
7534:^
7531:β
7501:β
7469:β
7453:−
7416:−
7358:β
7345:β
7318:β
7221:β
7208:β
7196:β
7166:β
7111:θ
7093:^
7090:θ
7071:^
7068:θ
7032:θ
7017:θ
7003:θ
6981:θ
6853:≥
6847:θ
6835:θ
6775:^
6772:θ
6727:∣
6721:^
6718:θ
6694:∣
6691:θ
6669:θ
6632:^
6629:θ
6599:^
6596:θ
6525:∣
6496:Λ
6493:⋅
6449:∣
6408:, is the
6320:posterior
6279:. By the
6228:∣
6219:θ
6197:∣
6188:θ
6162:∣
6153:θ
6140:θ
6133:Λ
6051:θ
5987:θ
5983:∂
5975:
5969:∂
5951:θ
5947:∂
5939:
5933:∂
5925:∞
5920:∞
5917:−
5913:∫
5903:θ
5863:∞
5854:≤
5816:∞
5811:∞
5808:−
5804:∫
5721:θ
5717:∂
5707:θ
5703:∂
5693:θ
5689:∂
5675:∂
5621:θ
5617:∂
5607:θ
5603:∂
5589:∂
5538:θ
5534:∂
5526:∂
5498:Θ
5495:∈
5492:θ
5440:…
5379:θ
5375:∂
5365:θ
5361:∂
5351:θ
5347:∂
5339:
5327:∂
5306:θ
5302:∂
5293:θ
5289:∂
5281:
5269:∂
5248:θ
5244:∂
5236:
5230:∂
5204:Θ
5201:∈
5198:θ
5137:Θ
5103:θ
5092:Θ
5089:∂
5086:→
5083:θ
5054:Θ
5051:∂
4983:θ
4979:∂
4970:∂
4957:≡
4951:∇
4929:Θ
4926:∈
4923:θ
4828:θ
4824:∂
4814:θ
4810:∂
4795:∂
4781:≡
4775:θ
4744:Θ
4741:∈
4735:^
4732:θ
4686:connected
4669:Θ
4652:concavity
4612:θ
4502:θ
4477:∣
4474:θ
4400:θ
4397:∣
4365:θ
4235:θ
4232:∣
4213:
4208:θ
4164:∣
4161:θ
4148:
4143:θ
4086:θ
4083:∣
4064:
4059:θ
4013:θ
4010:∣
3967:∫
3941:→
3925:
3920:θ
3838:∈
3832:∣
3829:θ
3804:→
3788:
3783:θ
3739:∣
3736:θ
3723:
3718:θ
3660:θ
3657:∣
3625:θ
3622:∣
3579:∫
3553:→
3506:θ
3503:∣
3460:∫
3446:
3441:θ
3372:∈
3366:∣
3363:θ
3350:
3345:θ
3295:θ
3292:∣
3248:θ
3245:∣
3202:∫
3188:
3183:θ
3149:θ
3146:∣
3127:≤
3121:≤
3092:
3087:θ
3018:∈
3012:∣
3009:θ
2986:
2981:θ
2875:∈
2869:∣
2866:θ
2843:
2838:θ
2769:∈
2763:∣
2760:θ
2747:
2742:θ
2659:∈
2653:∣
2650:θ
2477:θ
2457:θ
2384:θ
2372:, is the
2360:θ
2324:θ
2307:∣
2304:θ
2272:θ
2106:∣
2018:∣
1857:∣
1799:∣
1678:∣
1615:fair coin
1530:−
1417:θ
1391:∣
1388:θ
1356:∣
1353:θ
1323:θ
1279:θ
1198:θ
1103:, is the
1091:θ
1049:θ
1027:θ
1010:∣
1007:θ
975:θ
870:θ
839:θ
816:θ
781:θ
746:θ
743:∣
690:θ
670:θ
627:θ
624:∣
584:∣
581:θ
545:θ
542:∣
530:↦
527:θ
459:θ
456:∣
444:↦
412:θ
375:precision
340:parameter
334:explains
101:Coherence
55:Posterior
15420:Category
15113:Survival
14990:Johansen
14713:Binomial
14668:Isotonic
14255:(normal)
13900:location
13707:Blocking
13662:Sampling
13541:QâQ plot
13506:Box plot
13488:Graphics
13383:Skewness
13373:Kurtosis
13345:Variance
13275:Heronian
13270:Harmonic
13137:Statlect
12965:(1989).
12833:(eds.).
12502:(1922).
12434:(1999).
12432:Hald, A.
12149:25049882
11917:(1985).
11721:(1985),
11528:15896597
11399:(1995).
11196:See also
11095:post-hoc
11072:, where
10743:function
9641:and the
9194:for the
9061:→
8711:, where
8489:gradient
8429:(in the
6895:interval
6285:powerful
5040:boundary
2606:, where
1311:; it is
386:converse
67:Evidence
15446:Commons
15393:Kriging
15278:Process
15235:studies
15094:Wavelet
14927:General
14094:Plug-in
13888:L space
13667:Cluster
13368:Moments
13186:Outline
12598:Bibcode
12516:Bibcode
12488:: 3â32.
12464:2676741
12421:(2007).
12189:0400509
12122:2347496
11931:125â127
11809:(§2.1).
11727:(§9.3).
11565:2333005
11493:2240844
11020:improve
10710:is the
9716:
9685:
9677:
9646:
9639:
9617:
8740:is the
8652:of the
8230:of the
8174:
8154:
7879:profile
7807:is the
6972:vector
6647:. The
6406:
6386:
6382:
6355:
6351:
6324:
4756:if the
4647:compact
4634:assumed
2398:outcome
1453:Example
1109:outcome
346:of the
15472:Portal
15315:Census
14905:Normal
14853:Manova
14673:Robust
14423:2-way
14415:1-way
14253:-test
13924:
13501:Biplot
13292:Median
13285:Lehmer
13227:Center
13109:
13079:
13056:
13037:
12977:
12951:
12925:
12898:
12810:
12554:
12546:
12482:Metron
12462:
12268:
12212:
12187:
12147:
12120:
12081:
12053:
12026:
11995:
11966:
11937:
11751:
11665:
11586:
11563:
11526:
11491:
11446:
11379:
11354:
11329:
11304:
11009:, but
10963:given
10813:, and
9033:, and
8815:about
8811:in an
8748:, and
8487:, its
8485:smooth
8252:adds",
8226:, and
7991:events
7693:where
6258:: the
6005:
5999:
5963:
5774:where
5070:i.e.,
4520:where
4319:, the
3275:where
2928:since
2070:Hence
1655:i.i.d.
1211:
479:where
357:, the
14939:Trend
14468:prior
14410:anova
14299:-test
14273:-test
14265:-test
14172:Power
14117:Pivot
13910:shape
13905:scale
13355:Shape
13335:Range
13280:Heinz
13255:Cubic
13191:Index
12994:Risks
12556:91208
12552:JSTOR
12460:JSTOR
12145:JSTOR
12118:JSTOR
12020:21â24
11830:Wiley
11561:JSTOR
11524:S2CID
11489:JSTOR
11260:Notes
10876:(see
10606:Here
8493:score
8461:score
8423:graph
8417:Graph
7188:into
6961:graph
6410:prior
6074:score
4645:on a
3680:Then
2221:with
2213:be a
2120:0.09.
2056:0.09.
1813:0.25.
1716:0.25.
944:with
702:with
662:with
63:Prior
15172:Test
14372:Sign
14224:Wald
13297:Mode
13235:Mean
13107:ISBN
13077:ISBN
13054:ISBN
13035:ISBN
12975:ISBN
12949:ISBN
12923:ISBN
12896:ISBN
12808:ISBN
12266:ISBN
12210:ISBN
12079:ISBN
12051:ISBN
12024:ISBN
11993:ISBN
11964:ISBN
11935:ISBN
11749:ISBN
11663:ISBN
11584:ISBN
11444:ISBN
11377:ISBN
11352:ISBN
11327:ISBN
11302:ISBN
11272:See
11121:...
11065:...
10731:and
9762:and
9738:The
9490:and
9457:The
8471:and
8421:The
6816:for
6353:and
6316:odds
6029:and
5860:<
5737:<
5637:<
5554:<
4683:open
2617:>
2376:(of
2193:Let
1930:and
1872:0.25
920:Let
14352:BIC
14347:AIC
13012:hdl
13002:doi
12915:doi
12800:doi
12635:doi
12606:doi
12544:JFM
12534:hdl
12524:doi
12512:222
12450:doi
12380:doi
12351:doi
12324:doi
12297:doi
12177:doi
12110:doi
11694:doi
11640:doi
11613:doi
11553:doi
11516:doi
11479:doi
11186:AIC
11168:.
10864:In
10843:.
10408:log
10341:log
10253:log
10108:log
10035:log
9999:log
9987:log
9947:log
9425:log
9243:exp
9198:):
8807:is
8347:log
8322:log
8273:log
8258:or
8182:or
8152:or
7811:of
6885:If
6861:100
6651:of
6585:is
6551:In
6267:In
6025:is
5972:log
5936:log
5336:log
5278:log
5233:log
5079:lim
4910:is
4315:In
3934:lim
3797:lim
3546:lim
2103:0.3
2044:0.3
2034:0.3
1985:0.3
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912:).
860:not
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13101:.
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13029:.
13010:.
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12769:^
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12727:^
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12401:.
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12264:.
12260:.
12245:^
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12022:.
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11933:.
11921:.
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11839:^
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8095:.
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