707:
718:. In the point estimate we try to choose a unique point in the parameter space which can reasonably be considered as the true value of the parameter. On the other hand, instead of unique estimate of the parameter, we are interested in constructing a family of sets that contain the true (unknown) parameter value with a specified probability. In many problems of statistical inference we are not interested only in estimating the parameter or testing some hypothesis concerning the parameter, we also want to get a lower or an upper bound or both, for the real-valued parameter. To do this, we need to construct a confidence interval.
863:
471:) denote a random sample with joint p.d.f or p.m.f. f(x, θ) (θ may be a vector). The function f(x, θ), considered as a function of θ, is called the likelihood function. In this case, it is denoted by L(θ). The principle of maximum likelihood consists of choosing an estimate within the admissible range of θ, that maximizes the likelihood. This estimator is called the maximum likelihood estimate (MLE) of θ. In order to obtain the MLE of θ, we use the equation
851:) which we expect would include the true value of γ(θ). So this type of estimation is called confidence interval estimation. This estimation provides a range of values which the parameter is expected to lie. It generally gives more information than point estimates and are preferred when making inferences. In some way, we can say that point estimation is the opposite of interval estimation.
164:, the estimator T is called an unbiased estimator for the parameter θ if E = θ, irrespective of the value of θ. For example, from the same random sample we have E(x̄) = μ (mean) and E(s) = σ (variance), then x̄ and s would be unbiased estimators for μ and σ. The difference E − θ is called the bias of T ; if this difference is nonzero, then T is called biased.
456:, due to R.A. Fisher, is the most important general method of estimation. This estimator method attempts to acquire unknown parameters that maximize the likelihood function. It uses a known model (ex. the normal distribution) and uses the values of parameters in the model that maximize a likelihood function to find the most suitable match for the data.
501:, which uses all the known facts about a population and apply those facts to a sample of the population by deriving equations that relate the population moments to the unknown parameters. We can then solve with the sample mean of the population moments. However, due to the simplicity, this method is not always accurate and can be biased easily.
279:
the statistician would like to condense the data by computing some statistics and to base their analysis on these statistics so that there is no loss of relevant information in doing so, that is the statistician would like to choose those statistics which exhaust all information about the parameter, which is contained in the sample. We define
443:
Below are some commonly used methods of estimating unknown parameters which are expected to provide estimators having some of these important properties. In general, depending on the situation and the purpose of our study we apply any one of the methods that may be suitable among the methods of point
278:
In statistics, the job of a statistician is to interpret the data that they have collected and to draw statistically valid conclusion about the population under investigation. But in many cases the raw data, which are too numerous and too costly to store, are not suitable for this purpose. Therefore,
172:
Consistency is about whether the point estimate stays close to the value when the parameter increases its size. The larger the sample size, the more accurate the estimate is. If a point estimator is consistent, its expected value and variance should be close to the true value of the parameter. An
368:
The MAP estimator has good asymptotic properties, even for many difficult problems, on which the maximum-likelihood estimator has difficulties. For regular problems, where the maximum-likelihood estimator is consistent, the maximum-likelihood estimator ultimately agrees with the MAP estimator.
698:, also known as the Gauss–Markov theorem states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero.
239:. We can also say that the most efficient estimators are the ones with the least variability of outcomes. Therefore, if the estimator has smallest variance among sample to sample, it is both most efficient and unbiased. We extend the notion of efficiency by saying that estimator T
560:), r = 1, 2, …, k. In the method of moments, we equate k sample moments with the corresponding population moments. Generally, the first k moments are taken because the errors due to sampling increase with the order of the moment. Thus, we get k equations μ
812:. In general, with a normally-distributed sample mean, Ẋ, and with a known value for the standard deviation, σ, a 100(1-α)% confidence interval for the true μ is formed by taking Ẋ ± e, with e = z
134:. However, a biased estimator with a small variance may be more useful than an unbiased estimator with a large variance. Most importantly, we prefer point estimators that have the smallest
118:” is defined as the difference between the expected value of the estimator and the true value of the population parameter being estimated. It can also be described that the closer the
122:
of a parameter is to the measured parameter, the lesser the bias. When the estimated number and the true value is equal, the estimator is considered unbiased. This is called an
295:) be a random sample. A statistic T(X) is said to be sufficient for θ (or for the family of distribution) if the conditional distribution of X given T is free from θ.
979:
Statistical data analysis based on the L1-norm and related methods: Papers from the First
International Conference held at Neuchâtel, August 31–September 4, 1987
603:
In the method of least square, we consider the estimation of parameters using some specified form of the expectation and second moment of the observations. For
364:), which finds a maximum of the posterior distribution; for a uniform prior probability, the MAP estimator coincides with the maximum-likelihood estimator;
724:
describes how reliable an estimate is. We can calculate the upper and lower confidence limits of the intervals from the observed data. Suppose a dataset x
648:(BLUE). Again, if we assume that the least square estimates are independently and identically normally distributed, then a linear estimator will be
1100:
820:
is the 100(1-α/2)% cumulative value of the standard normal curve, and n is the number of data values in that column. For example, z
497:
was introduced by K. Pearson and P. Chebyshev in 1887, and it is one of the oldest methods of estimation. This method is based on
262:
Generally, we must consider the distribution of the population when determining the efficiency of estimators. For example, in a
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and it is claimed with a certain degree of confidence (measured in probabilistic terms) that the true value of γ lies between l
1012:
665:
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347:
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485:=0, i = 1, 2, …, k. If θ is a vector, then partial derivatives are considered to get the likelihood equations.
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453:
266:, the mean is considered more efficient than the median, but the same does not apply in asymmetrical, or
740:. Let θ be the parameter of interest, and γ a number between 0 and 1. If there exist sample statistics L
1249:
1004:
653:
706:
684:
644:) is a linear function of the parameters and the x-values are known, least square estimators will be
340:; in Bayesian estimation, the risk is defined in terms of the posterior distribution, as observed by
334:
982:
901:
580:, r = 1, 2, …, k. Solving these equations we get the method of moment estimators (or estimates) as
431:
419:
94:
81:. More generally, a point estimator can be contrasted with a set estimator. Examples are given by
886:
90:
540:. Further, let the first k population moments about zero exist as explicit function of θ, i.e. μ
324:
891:
626:), i = 1, 2,…n, we may use the method of least squares. This method consists of minimizing the
389:
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A point estimator can also be contrasted with a distribution estimator. Examples are given by
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115:
350:, which minimizes the posterior risk for the absolute-value loss function, as observed by
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unbiased estimator is consistent if the limit of the variance of estimator T equals zero.
8:
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Experimental Design – With
Applications in Management, Engineering, and the Sciences
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46:) which is to serve as a "best guess" or "best estimate" of an unknown population
1071:
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51:
43:
1226:. Vol. I (Second (updated printing 2007) ed.). Pearson Prentice-Hall.
119:
516:) be a random sample from a population having p.d.f. (or p.m.f) f(x,θ), θ = (θ
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1208:. Springer: Paul D. Berger, Robert E. Maurer, Giovana B. Celli. 2019.
55:
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Here two limits are computed from the set of observations, say l
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Statistical
Decision Theory: Estimation, Testing, and Selection
937:. F.M. Dekking, C. Kraaikamp, H.P. Lopuhaa, L.E. Meester. 2005.
652:(MVUE) for the entire class of unbiased estimators. See also
341:
960:. Pradip Kumar Sahu, Santi Ranjan Pal, Ajit Kumar Das. 2015.
714:
There are two major types of estimates: point estimate and
31:
690:
687:
minimizes the risk of the absolute-error loss function.
1098:(1982). "An inconsistent maximum likelihood estimate".
732:
is given, modeled as realization of random variables X
1194:. New York: John Wiley & Sons: Theil Henri. 1971.
447:
438:
858:
710:
Point estimation and confidence interval estimation.
935:
54:). More formally, it is the application of a point
1224:Mathematical Statistics: Basic and Selected Topics
195:be two unbiased estimators for the same parameter
1231:Liese, Friedrich & Miescke, Klaus-J. (2008).
1166:. John Wiley and Sons, New York: Agresti A. 1990.
1068:Asymptotic Methods in Statistical Decision Theory
247:(for the same parameter of interest), if the MSE(
1241:
1222:Bickel, Peter J. & Doksum, Kjell A. (2001).
702:Point estimate v.s. confidence interval estimate
528:). The objective is to estimate the parameters θ
104:
1101:Journal of the American Statistical Association
660:Minimum-variance mean-unbiased estimator (MVUE)
422:have close connections with Bayesian analysis:
316:are the posterior distribution's statistics of
65:: such interval estimates are typically either
1192:Best Linear Unbiased Estimation and Prediction
1134:
1128:
308:Bayesian inference is typically based on the
679:
303:
298:
152:) be an estimator based on a random sample X
488:
16:Parameter estimation via sample statistics
598:
1094:
1027:
1001:Probability Theory: The logic of science
705:
61:Point estimation can be contrasted with
58:to the data to obtain a point estimate.
34:to calculate a single value (known as a
1088:
1058:
1056:
969:
967:
606:fitting a curve of the form y = f( x, β
384:) point estimator is based in Bayesian
1242:
1178:The Concise Encyclopedia of Statistics
1062:
995:
388:and is not so directly related to the
973:
958:Estimation and Inferential Statistics
691:Best linear unbiased estimator (BLUE)
672:(expected loss) of the squared-error
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772:) = γ for every value of θ, then (l
666:minimum-variance unbiased estimator
650:minimum-variance unbiased estimator
320:, e.g., its mean, median, or mode:
13:
1215:
808:for θ. The number γ is called the
448:Method of maximum likelihood (MLE)
439:Methods of finding point estimates
327:, which minimizes the (posterior)
243:is more efficient than estimator T
14:
1261:
941:
918:
861:
824:equals 1.96 for 95% confidence.
235:), irrespective of the value of
1198:
1184:
1170:
1033:A Course in Large Sample Theory
1156:
1114:10.1080/01621459.1982.10477894
1021:
989:
696:Best linear unbiased estimator
646:best linear unbiased estimator
273:
167:
1:
912:
593:generalized method of moments
176:
109:
105:Properties of point estimates
843:. Thus we get an interval (l
716:confidence interval estimate
454:method of maximum likelihood
255:is smaller than the MSE of T
126:The estimator will become a
7:
1180:. Springer: Dodge, Y. 2008.
854:
10:
1266:
1142:(2nd ed.). Springer.
1140:Theory of Point Estimation
1005:Cambridge University Press
654:minimum mean squared error
1164:Categorical Data Analysis
685:Median-unbiased estimator
680:Median unbiased estimator
314:Bayesian point estimators
304:Bayesian point estimation
299:Types of point estimation
1096:Ferguson, Thomas S.
983:North-Holland Publishing
902:Philosophy of statistics
432:Markov chain Monte Carlo
420:computational statistics
369:Bayesian estimators are
91:confidence distributions
887:Confidence distribution
489:Method of moments (MOM)
128:best unbiased estimator
1138:; Casella, G. (1998).
1003:(5. print. ed.).
892:Induction (philosophy)
711:
599:Method of least square
390:posterior distribution
378:Minimum Message Length
333:(expected loss) for a
310:posterior distribution
283:as follows: Let X =( X
38:since it identifies a
882:Binomial distribution
877:Algorithmic inference
804:), is called a 100γ%
709:
373:, by Wald's theorem.
281:sufficient statistics
95:randomized estimators
71:frequentist inference
907:Predictive inference
499:law of large numbers
358:maximum a posteriori
67:confidence intervals
27:involves the use of
1029:Ferguson, Thomas S.
897:Interval estimation
806:confidence interval
722:Confidence interval
264:normal distribution
136:mean square errors.
124:unbiased estimator.
99:Bayesian posteriors
63:interval estimation
1037:Chapman & Hall
869:Mathematics portal
712:
386:information theory
130:if it has minimum
79:Bayesian inference
75:credible intervals
50:(for example, the
1250:Estimation theory
1014:978-0-521-59271-0
495:method of moments
395:Special cases of
270:, distributions.
249:mean square error
140:If we let T = h(X
77:, in the case of
69:, in the case of
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618:) to the data (x
397:Bayesian filters
348:Posterior median
318:central tendency
206:would be called
199:. The estimator
25:point estimation
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977:, ed. (1987).
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1235:. Springer.
800:, . . . , x
788:, . . . , x
760:, . . . , X
748:, . . . , X
736:, . . . , X
728:, . . . , x
632:When f(x, β
591:. See also
274:Sufficiency
168:Consistency
160:, . . . , X
148:, . . . , X
913:References
780:), where l
459:Let X = (X
371:admissible
177:Efficiency
110:Biasedness
21:statistics
56:estimator
48:parameter
1244:Category
1066:(1986).
1031:(1996).
999:(2007).
855:See also
656:(MMSE).
640:, ,,,, β
614:, ,,,, β
587:= 1/n ΣX
536:, ..., θ
467:, ... ,X
414:Several
291:, ... ,X
132:variance
42:in some
1122:2287314
792:) and u
752:) and U
416:methods
352:Laplace
312:. Many
217:if Var(
1146:
1120:
1078:
1043:
1011:
524:, …, θ
504:Let (X
434:(MCMC)
268:skewed
251:) of T
97:, and
29:sample
1118:JSTOR
847:and u
839:and u
831:and u
822:1-α/2
818:1-α/2
814:1-α/2
796:= h(x
784:= g(x
756:= h(X
744:= g(X
576:) = m
572:,…, θ
556:,…, θ
477:L(θ)/
342:Gauss
73:, or
40:point
1144:ISBN
1076:ISBN
1041:ISBN
1009:ISBN
670:risk
493:The
475:dlog
452:The
376:The
330:risk
228:Var(
226:<
188:and
181:Let
116:Bias
32:data
1110:doi
776:, u
636:, β
622:, y
610:, β
568:, θ
552:, θ
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532:, θ
520:, θ
512:,…X
508:, X
463:, X
418:of
382:MML
362:MAP
287:, X
85:or
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