108:(1937) introduced the idea of "confidence" in his seminal paper on confidence intervals which clarified the frequentist repetition property. According to Fraser, the seed (idea) of confidence distribution can even be traced back to Bayes (1763) and Fisher (1930). Although the phrase seems to first be used in Cox (1958). Some researchers view the confidence distribution as "the Neymanian interpretation of Fisher's fiducial distributions", which was "furiously disputed by Fisher". It is also believed that these "unproductive disputes" and Fisher's "stubborn insistence" might be the reason that the concept of confidence distribution has been long misconstrued as a fiducial concept and not been fully developed under the frequentist framework. Indeed, the confidence distribution is a purely frequentist concept with a purely frequentist interpretation, although it also has ties to Bayesian and fiducial inference concepts.
2982:
3656:
3291:
35:) has often been loosely referred to as a distribution function on the parameter space that can represent confidence intervals of all levels for a parameter of interest. Historically, it has typically been constructed by inverting the upper limits of lower sided confidence intervals of all levels, and it was also commonly associated with a fiducial interpretation (
2659:
1370:
468:
that inferences (point estimators, confidence intervals and hypothesis testing, etc.) based on the confidence distribution have desired frequentist properties. This is similar to the restrictions in point estimation to ensure certain desired properties, such as unbiasedness, consistency, efficiency, etc.
2638:
476:
there is a unique "best" (in terms of optimality) confidence distribution. But sometimes there is no optimal confidence distribution available or, in some extreme cases, we may not even be able to find a meaningful confidence distribution. This is not different from the practice of point estimation.
61:
distribution. The development and interpretation of a bootstrap distribution does not involve any fiducial reasoning; the same is true for the concept of a confidence distribution. But the notion of confidence distribution is much broader than that of a bootstrap distribution. In particular, recent
467:
In nontechnical terms, a confidence distribution is a function of both the parameter and the random sample, with two requirements. The first requirement (R1) simply requires that a CD should be a distribution on the parameter space. The second requirement (R2) sets a restriction on the function so
3056:
475:
Unlike the classical fiducial inference, more than one confidence distributions may be available to estimate a parameter under any specific setting. Also, unlike the classical fiducial inference, optimality is not a part of requirement. Depending on the setting and the criterion used, sometimes
225:
To interpret the CD function entirely from a frequentist viewpoint and not interpret it as a distribution function of a (fixed/nonrandom) parameter is one of the major departures of recent development relative to the classical approach. The nice thing about treating confidence distributions as a
2977:{\displaystyle \pi (\rho |r)={\frac {\nu (\nu -1)\Gamma (\nu -1)}{{\sqrt {2\pi }}\Gamma (\nu +{\frac {1}{2}})}}(1-r^{2})^{\frac {\nu -1}{2}}\cdot (1-\rho ^{2})^{\frac {\nu -2}{2}}\cdot (1-r\rho )^{\frac {1-2\nu }{2}}F({\frac {3}{2}},-{\frac {1}{2}};\nu +{\frac {1}{2}};{\frac {1+r\rho }{2}})}
1175:
2143:
4407:
471:
A confidence distribution derived by inverting the upper limits of confidence intervals (classical definition) also satisfies the requirements in the above definition and this version of the definition is consistent with the classical definition.
1474:
2257:
4154:. The same holds for a CD, where the confidence level is achieved in limit. Some authors have proposed using them for graphically viewing what parameter values are consistent with the data, instead of coverage or performance purposes.
2471:
4280:
3286:{\displaystyle \pi (\rho |r)={\frac {(1-r^{2})^{\frac {\nu -1}{2}}\cdot (1-\rho ^{2})^{\frac {\nu -2}{2}}}{\pi (\nu -2)!}}\partial _{\rho r}^{\nu -2}\left\{{\frac {\theta -{\frac {1}{2}}\sin 2\theta }{\sin ^{3}\theta }}\right\}}
3618:, and the confidence regions can be chosen in many other ways. The confidence distribution coincides in this case with the Bayesian posterior using the right Haar prior. The argument generalizes to the case of an unknown mean
1950:
62:
research suggests that it encompasses and unifies a wide range of examples, from regular parametric cases (including most examples of the classical development of Fisher's fiducial distribution) to bootstrap distributions,
2452:
1365:{\displaystyle H_{\Phi }(\mu )={\mathit {\Phi }}\left({\frac {{\sqrt {n}}(\mu -{\bar {X}})}{\sigma }}\right),\quad {\text{and}}\quad H_{t}(\mu )=F_{t_{n-1}}\left({\frac {{\sqrt {n}}(\mu -{\bar {X}})}{s}}\right),}
4530:
39:), although it is a purely frequentist concept. A confidence distribution is NOT a probability distribution function of the parameter of interest, but may still be a function useful for making inferences.
3767:
2357:
6206:
Contemporary
Developments in Bayesian Analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman. (D. Fourdrinier, et al., Eds.). IMS Collection, Volume 8, 200 -214.
4412:
Under some modest conditions, among other properties, one can prove that these point estimators are all consistent. Certain confidence distributions can give optimal frequentist estimators.
2052:
226:
purely frequentist concept (similar to a point estimator) is that it is now free from those restrictive, if not controversial, constraints set forth by Fisher on fiducial distributions.
42:
In recent years, there has been a surge of renewed interest in confidence distributions. In the more recent developments, the concept of confidence distribution has emerged as a purely
1733:
1608:
1131:
938:
4152:
5601:
3363:
3331:
2002:
1659:
748:
4288:
4066:
3494:
2040:
6123:
5142:
4106:
5329:
Singh, K. Xie, M. and
Strawderman, W.E. (2001). "Confidence distributions—concept, theory and applications". Technical report, Dept. Statistics, Rutgers Univ. Revised 2004.
3042:
5948:
concurve: Computes and Plots
Compatibility (Confidence) Intervals, P-Values, S-Values, & Likelihood Intervals to Form Consonance, Surprisal, & Likelihood Functions
5767:
3596:
3407:
1817:
1777:
1557:
1521:
1167:
1060:
677:
1385:
575:
218:." and "it has powerful intuitive appeal". In the classical literature, the confidence distribution function is interpreted as a distribution function of the parameter
3636:
3616:
3565:
3514:
3427:
1093:
958:
625:
525:
3541:
886:
835:
704:
605:
2152:
1679:
3451:
3004:
2633:{\displaystyle H_{n}(\rho )=1-{\mathit {\Phi }}\left({\sqrt {n-3}}\left({1 \over 2}\ln {1+r \over 1-r}-{1 \over 2}\ln {{1+\rho } \over {1-\rho }}\right)\right)}
855:
808:
788:
768:
645:
502:
3976:
3864:
5967:
5265:
Singh, K. and Xie, M. (2011). "Discussions of “Is Bayes posterior just quick and dirty confidence?” by D.A.S. Fraser." Statistical
Science. Vol. 26, 319-321.
4200:
54:), but it uses a sample-dependent distribution function on the parameter space (instead of a point or an interval) to estimate the parameter of interest.
1836:
3665:
97:
and p-values, among others. Some recent developments have highlighted the promising potentials of the CD concept, as an effective inferential tool.
5694:
4998:
2368:
121:
Classically, a confidence distribution is defined by inverting the upper limits of a series of lower-sided confidence intervals. In particular,
6203:
2276:
5968:"Concurve plots consonance curves, p-value functions, and S-value functions « Statistical Modeling, Causal Inference, and Social Science"
5037:
Xie, M. and Singh, K. (2013). "Confidence
Distribution, the Frequentist Distribution Estimator of a Parameter – a Review (with discussion)".
5430:"Nonparametric Fusion Learning for Multiparameters: Synthesize Inferences From Diverse Sources Using Data Depth and Confidence Distribution"
1739:
and it violates the two requirements in the CD definition, it is no longer a "distribution estimator" or a confidence distribution for
4444:
4162:
Point estimators can also be constructed given a confidence distribution estimator for the parameter of interest. For example, given
222:, which is impossible unless fiducial reasoning is involved since, in a frequentist setting, the parameters are fixed and nonrandom.
85:, a confidence distribution contains a wealth of information for constructing almost all types of frequentist inferences, including
6115:
5341:
Singh, K. Xie, M. and
Strawderman, W.E. (2005). "Combining Information from Independent Sources Through Confidence Distribution"
5844:
4919:
5862:"Semantic and cognitive tools to aid statistical science: replace confidence and significance by compatibility and surprise"
4949:
Xie, M. (2013). "Rejoinder of
Confidence Distribution, the Frequentist Distribution Estimator of a Parameter – a Review".
5994:
pvaluefunctions: Creates and Plots P-Value
Functions, S-Value Functions, Confidence Distributions and Confidence Densities
46:
concept, without any fiducial interpretation or reasoning. Conceptually, a confidence distribution is no different from a
5479:
Singh, K. Xie, M. and
Strawderman, W.E. (2007). "Confidence Distribution (CD)-Distribution Estimator of a Parameter", in
2138:{\displaystyle H_{\mathit {\Phi }}(\mu )={\mathit {\Phi }}\left({\frac {{\sqrt {n}}(\mu -{\bar {X}})}{\sigma }}\right)}
6181:
2297:
6107:
6093:
5670:
5293:
3044:. This is also the posterior density of a Bayes matching prior for the five parameters in the binormal distribution.
2145:
is optimal in terms of producing the shortest confidence intervals at any given level. In the case when the variance
6119:
Ser. B. 17, 69—78. (criticism of statistical theories of Jerzy Neyman and
Abraham Wald from a fiducial perspective)
3454:
2280:
5102:
Neyman, J. (1937). "Outline of a theory of statistical estimation based on the classical theory of probability."
6225:
1700:
1562:
1098:
895:
4541:(C) is called "support" in the CD inference and also known as "belief" in the fiducial literature. We have
4402:{\displaystyle {\widehat {\theta }}_{n}=\arg \max _{\theta }h_{n}(\theta ),h_{n}(\theta )=H'_{n}(\theta ).}
4115:
1375:
satisfy the two requirements in the CD definition, and they are confidence distribution functions for
5368:
3336:
3298:
960:
is a real parameter, then the measure theoretic definition coincides with the above classical definition.
6152:
4832:
A few statistical programs have implemented the ability to construct and graph confidence distributions.
1962:
1613:
709:
4027:
57:
A simple example of a confidence distribution, that has been broadly used in statistical practice, is a
6220:
58:
6035:
6182:
Population Estimates From Aerial Photographic Surveys of Naturally and Variably Marked Bowhead Whales
5429:
3460:
2007:
5586:
5411:
4075:
6138:
4835:
3009:
1469:{\displaystyle H_{A}(\mu )={\mathit {\Phi }}\left({\frac {{\sqrt {n}}(\mu -{\bar {X}})}{s}}\right)}
4420:
One can derive a p-value for a test, either one-sided or two-sided, concerning the parameter
3574:
3380:
2363:
1786:
1746:
1526:
1490:
1136:
1025:
650:
940:
holds with equality, then the confidence distribution is by definition exact. If, additionally,
538:
75:
6069:
5573:
5398:
3621:
3601:
3550:
3499:
3412:
2288:
2252:{\displaystyle H_{t}(\mu )=F_{t_{n-1}}\left({\frac {{\sqrt {n}}(\mu -{\bar {X}})}{s}}\right)}
1065:
943:
610:
510:
505:
43:
36:
24:
6071:"Confidence Distribution, the Frequentist Distribution Estimator of a Parameter: A Review".
5343:
4877:
3519:
864:
813:
682:
583:
81:
Just as a Bayesian posterior distribution contains a wealth of information for any type of
1664:
8:
4824:
See Figure 1 from Xie and Singh (2011) for a graphical illustration of the CD inference.
974:
90:
67:
51:
17:
6086:
Confidence, Likelihood, Probability: Statistical Inference with Confidence Distributions
6170:
5906:
5873:
5861:
5807:
5779:
5745:
5737:
5688:
5641:
5613:
5538:
5491:
5457:
5352:
5266:
5126:
5007:
4981:
3436:
2989:
889:
840:
793:
773:
753:
630:
487:
82:
71:
4275:{\displaystyle {\bar {\theta }}_{n}=\int _{-\infty }^{\infty }t\,\mathrm {d} H_{n}(t)}
3881:
6103:
6089:
5992:
5911:
5893:
5840:
5799:
5749:
5729:
5676:
5666:
5633:
5542:
5530:
5461:
5449:
5289:
5199:
5085:
3568:
578:
94:
5826:
5811:
5645:
5901:
5883:
5832:
5789:
5721:
5623:
5561:
5520:
5441:
5386:
5227:
5189:
5075:
5046:
4958:
4923:
3430:
528:
6192:
5445:
5177:
5080:
5063:
4853:
1022:
be the cumulative distribution function of the standard normal distribution, and
47:
5565:
5390:
1945:{\displaystyle H_{\chi ^{2}}(\theta )=1-F_{\chi _{n-1}^{2}}((n-1)s^{2}/\theta )}
5888:
5557:
5525:
5382:
5369:"Incorporating expert opinions with information from binomial clinical trials."
5308:
Efron, B. (1993). "Bayes and likelihood calculations from confidence intervals.
858:
86:
6014:
5946:
5725:
5194:
4927:
3762:{\displaystyle (-\infty ,H_{n}^{-1}(1-\alpha )],[H_{n}^{-1}(\alpha ),\infty )}
6214:
6204:"CD-posterior --- combining prior and data through confidence distributions."
6153:
The Fisher, Neyman–Pearson theories of testing hypotheses: one theory or two?
5897:
5803:
5733:
5637:
5534:
5508:
5453:
5203:
5118:
Fraser, D.A.S. (2011). "Is Bayes posterior just quick and dirty confidence?"
5089:
4899:
3639:
3048:
5231:
4996:
Fraser, D.A.S. (1991). "Statistical inference: Likelihood to significance."
3642:, but in this case the confidence distribution is not a Bayesian posterior.
214:
lying between the upper endpoints of the 0.90 and 0.95 confidence interval,
6177:. (reply to Fisher 1955, which diagnoses a fallacy of "fiducial inference")
6083:
5915:
5281:
2447:{\displaystyle N({1 \over 2}\ln {{1+\rho } \over {1-\rho }},{1 \over n-3})}
105:
5680:
455:
requirement is true only asymptotically and the continuity requirement on
4918:
Cox, D.R. (1958). "Some Problems Connected with Statistical Inference", "
5741:
5709:
5495:
5270:
5130:
6174:
5794:
5660:
5356:
5311:
5159:
5050:
5011:
4985:
4962:
1479:
satisfies the definition of an asymptotic confidence distribution when
314: → is called a confidence distribution (CD) for a parameter
5628:
6113:
Fisher, R. A. (1955). "Statistical methods and scientific induction"
3366:
532:
5878:
5836:
5784:
5618:
5219:
Schweder, T. and Hjort, N.L. (2002). "Confidence and likelihood",
4525:{\displaystyle p_{s}(C)=H_{n}(C)=\int _{C}\mathrm {d} H(\theta ).}
6165:
Neyman, Jerzy (1956). "Note on an Article by Sir Ronald Fisher".
3598:. The confidence distribution is in this case binormal with mean
3544:
210:
Efron stated that this distribution "assigns probability 0.05 to
63:
6191:
Bityukov S., Krasnikov N., Nadarajah S. and Smirnova V. (2010) "
6186:
Journal of Agricultural Biological and Environmental Statistics
3655:
3645:
361:, •) is a continuous cumulative distribution function on
6139:
Frequentist prediction intervals and predictive distributions
5143:
An Essay Towards Solving a Problem in the Doctrine of Chances
4862:
238:
is the parameter space of the unknown parameter of interest
6043:
5714:
Sankhyā: The Indian Journal of Statistics, Series A (2008-)
6013:
Black, James; Rothman, Ken; Thelwall, Simon (2019-01-23),
2266:
5245:
Zabell, S.L. (1992). "R.A.Fisher and fiducial argument",
4179:, natural choices of point estimators include the median
3457:
and known distribution in the plane. The distribution of
5944:
5428:
Liu, Dungang; Liu, Regina Y.; Xie, Min-ge (2021-04-30).
3662:
From the CD definition, it is evident that the interval
1830:, the sample-dependent cumulative distribution function
1483:→∞, and it is an asymptotic confidence distribution for
837:
are measurable functions of the data. This implies that
6180:
Schweder T., Sadykova D., Rugh D. and Koski W. (2010) "
5945:
Rafi [aut, Zad; cre; Vigotsky, Andrew D. (2020-04-20),
4586:, ∞), one can show from the CD definition that sup
3870:)%-level confidence intervals of different kinds, for
968:
479:
175:,α) is continuous and increasing in α for each sample
5940:
5938:
5766:
Taraldsen, Gunnar; Lindqvist, Bo Henry (2013-02-01).
5710:"Invariance, model matching and probability matching"
4447:
4291:
4203:
4118:
4078:
4030:
3884:
3770:
3668:
3624:
3604:
3577:
3553:
3522:
3502:
3463:
3439:
3415:
3383:
3339:
3301:
3059:
3012:
2992:
2662:
2474:
2371:
2300:
2155:
2055:
2010:
1965:
1839:
1789:
1749:
1703:
1667:
1616:
1565:
1529:
1493:
1388:
1178:
1139:
1101:
1068:
1028:
946:
898:
867:
843:
816:
796:
776:
756:
712:
685:
679:. The family of confidence regions is not unique. If
653:
633:
613:
586:
541:
513:
490:
6012:
5367:
Xie, M., Liu, R., Daramuju, C.V., Olsan, W. (2012).
5178:"Some Problems Connected with Statistical Inference"
1062:
the cumulative distribution function of the Student
4973:Efron, B. (1998). "R.A.Fisher in the 21st Century"
4799:)} is the corresponding p-value of the test. Here,
5935:
5765:
4524:
4401:
4274:
4146:
4100:
4060:
3970:
3858:
3761:
3630:
3610:
3590:
3559:
3535:
3508:
3488:
3445:
3421:
3401:
3357:
3325:
3285:
3036:
2998:
2976:
2632:
2446:
2351:
2251:
2137:
2034:
1996:
1944:
1811:
1771:
1727:
1673:
1653:
1602:
1551:
1515:
1468:
1364:
1161:
1125:
1087:
1054:
952:
932:
880:
849:
829:
802:
782:
762:
742:
698:
671:
639:
619:
599:
569:
519:
496:
6193:Confidence distributions in statistical inference
5602:"Objective priors for the bivariate normal model"
142:)] be a 100α% lower-side confidence interval for
6212:
5064:"The p-value Function and Statistical Inference"
4321:
2352:{\displaystyle z={1 \over 2}\ln {1+r \over 1-r}}
6157:Journal of the American Statistical Association
5708:Eaton, Morris L.; Sudderth, William D. (2012).
5434:Journal of the American Statistical Association
5215:
5213:
4999:Journal of the American Statistical Association
3047:The very last formula in the classical book by
2004:is the cumulative distribution function of the
6169:. Series B (Methodological) 18 (2): 288–294.
5859:
5707:
5600:Berger, James O.; Sun, Dongchu (2008-04-01).
5241:
5239:
4722:), one can show from the CD definition that p
2643:is an asymptotic confidence distribution for
6100:Statistical Methods and Scientific Inference
5761:
5759:
5693:: CS1 maint: multiple names: authors list (
5662:Statistical methods and scientific inference
5475:
5473:
5471:
5337:
5335:
5304:
5302:
5261:
5259:
5210:
4895:
4893:
4651:) is the corresponding p-value of the test.
3646:Using confidence distributions for inference
3006:is the Gaussian hypergeometric function and
770:is a confidence distribution with level set
5922:
5860:Rafi, Zad; Greenland, Sander (2020-09-30).
5423:
5421:
4441:under the confidence distribution function
4437:). Denote by the probability mass of a set
3372:
2283:population. It is well known that Fisher's
5824:
5236:
4945:
4943:
4941:
4939:
4937:
4935:
4282:, and the maximum point of the CD density
3567:and axes given by the eigenvectors of the
2259:is an optimal confidence distribution for
1955:is a confidence distribution function for
246:is the sample space corresponding to data
202:(•) is a confidence distribution for
5905:
5887:
5877:
5825:Cox, D. R.; Hinkley, D. V. (1979-09-06).
5793:
5783:
5756:
5627:
5617:
5599:
5555:
5524:
5506:
5468:
5427:
5380:
5332:
5323:
5299:
5256:
5193:
5135:
5112:
5096:
5079:
5033:
5031:
5029:
5027:
5025:
5023:
5021:
5019:
4990:
4890:
4247:
3992:)% confidence interval for the parameter
1728:{\displaystyle H_{\mathit {\Phi }}(\mu )}
1603:{\displaystyle N({\bar {X}},\sigma ^{2})}
1126:{\displaystyle H_{\mathit {\Phi }}(\mu )}
933:{\displaystyle P(\gamma \in A_{p})\geq p}
6167:Journal of the Royal Statistical Society
5990:
5509:"The Confidence Density for Correlation"
5418:
5275:
4967:
3654:
229:
5768:"Fiducial theory and optimal inference"
5371:Annals of Applied Statistics. In press.
5361:
4932:
4922:", "29" 357-372 (Section 4, Page 363)
2454:with a fast rate of convergence, where
2267:Example 2: Bivariate normal correlation
1002: = 1, 2, ...,
116:
6213:
6137:Lawless, F. and Fredette, M. (2005). "
6088:. London: Cambridge University Press.
5928:Kendall, M., & Stuart, A. (1974).
5658:
5061:
5016:
4912:
3650:
3496:defines a confidence distribution for
5932:, Volume ?. (Chapter 21). Wiley.
5665:( ed.). New York: Hafner Press.
5481:Complex Datasets and Inverse Problems
5182:The Annals of Mathematical Statistics
4920:The Annals of Mathematical Statistics
4415:
4147:{\displaystyle H_{n}(\theta )=\beta }
6082:Schweder, T and Hjort, N L (2016).
3358:{\displaystyle 0<\theta <\pi }
3326:{\displaystyle \cos \theta =-\rho r}
1559:are equivalent to state that we use
5659:Fisher, Ronald Aylmer, Sir (1973).
5286:Principles of Statistical Inference
5221:Scandinavian Journal of Statistics.
5175:
4424:, from its confidence distribution
4157:
3978:is a level 100(1 −
1997:{\displaystyle F_{\chi _{n-1}^{2}}}
1654:{\displaystyle N({\bar {X}},s^{2})}
969:Example 1: Normal mean and variance
743:{\displaystyle p\in I\subset (0,1)}
480:A definition with measurable spaces
433:, follows the uniform distribution
13:
6036:"Modern Epidemiology, 2nd Edition"
5484:IMS Lecture Notes—Monograph Series
4827:
4578:is of the type of (−∞,
4503:
4249:
4239:
4234:
4061:{\displaystyle H_{n}^{-1}(\beta )}
3753:
3675:
3579:
3465:
3200:
2737:
2707:
2505:
2082:
2062:
1710:
1413:
1203:
1184:
1108:
318:, if it follows two requirements:
234:The following definition applies;
14:
6237:
6124:On generalized fiducial inference
5930:The Advanced Theory of Statistics
3543:can be chosen as the interior of
1095:distribution. Both the functions
368:(R2) At the true parameter value
6073:International Statistical Review
5866:BMC Medical Research Methodology
5490:, (R. Liu, et al. Eds) 132–150.
5383:"Joint Confidence Distributions"
5039:International Statistical Review
4951:International Statistical Review
3878: ∈ (0, 1). Also
3866:provide 100(1 −
2650:An exact confidence density for
6195:". AIP Conference Proceedings,
6061:
6028:
6006:
5984:
5960:
5853:
5818:
5701:
5652:
5593:
5549:
5500:
5374:
5169:
5062:Fraser, D. A. S. (2019-03-29).
4902:(1930). "Inverse probability."
3489:{\displaystyle \Gamma ^{y}=y-U}
2035:{\displaystyle \chi _{n-1}^{2}}
1735:involves the unknown parameter
1265:
1259:
422:), as a function of the sample
6202:Singh, K. and Xie, M. (2012).
6068:Xie, M. and Singh, K. (2013).
5991:Infanger, Denis (2019-11-29),
5972:statmodeling.stat.columbia.edu
5055:
4516:
4510:
4486:
4480:
4464:
4458:
4393:
4387:
4368:
4362:
4346:
4340:
4269:
4263:
4211:
4135:
4129:
4101:{\displaystyle H_{n}(\theta )}
4095:
4089:
4055:
4049:
3965:
3962:
3943:
3919:
3906:
3885:
3853:
3850:
3830:
3806:
3792:
3771:
3756:
3747:
3741:
3720:
3714:
3711:
3699:
3669:
3365:. This formula was derived by
3190:
3178:
3151:
3131:
3106:
3086:
3077:
3070:
3063:
2971:
2899:
2871:
2855:
2830:
2810:
2785:
2765:
2759:
2740:
2722:
2710:
2704:
2692:
2680:
2673:
2666:
2491:
2485:
2458:is the sample correlation and
2441:
2375:
2236:
2230:
2215:
2172:
2166:
2122:
2116:
2101:
2074:
2068:
2045:In the case when the variance
1939:
1918:
1906:
1903:
1863:
1857:
1806:
1800:
1766:
1760:
1722:
1716:
1648:
1629:
1620:
1597:
1578:
1569:
1546:
1540:
1510:
1504:
1453:
1447:
1432:
1405:
1399:
1346:
1340:
1325:
1282:
1276:
1243:
1237:
1222:
1195:
1189:
1156:
1150:
1120:
1114:
921:
902:
737:
725:
558:
545:
129:in (0, 1), let (−∞,
1:
5446:10.1080/01621459.2021.1902817
5081:10.1080/00031305.2018.1556735
4883:
3037:{\displaystyle \nu =n-1>1}
111:
70:and, in some cases, Bayesian
6016:episheet: Rothman's Episheet
3429:is an unknown vector in the
892:If the defining requirement
7:
5566:10.13140/RG.2.2.23673.49769
5558:"Confidence in Correlation"
5391:10.13140/RG.2.2.33079.85920
4904:Proc. cambridge Pilos. Soc.
4871:
4654:(2) For the singleton test
4544:(1) For the one-sided test
3638:in an infinite-dimensional
3591:{\displaystyle \Gamma ^{y}}
3402:{\displaystyle Y=\gamma +U}
1812:{\displaystyle H_{A}(\mu )}
1772:{\displaystyle H_{t}(\mu )}
1552:{\displaystyle H_{A}(\mu )}
1516:{\displaystyle H_{t}(\mu )}
1162:{\displaystyle H_{t}(\mu )}
1055:{\displaystyle F_{t_{n-1}}}
963:
672:{\displaystyle 0<p<1}
10:
6242:
5889:10.1186/s12874-020-01105-9
5556:Taraldsen, Gunnar (2020).
5526:10.1007/s13171-021-00267-y
5507:Taraldsen, Gunnar (2021).
5381:Taraldsen, Gunnar (2021).
570:{\displaystyle C(A_{p})=p}
484:A confidence distribution
100:
50:or an interval estimator (
15:
5726:10.1007/s13171-012-0018-4
5068:The American Statistician
4175:) the CD for a parameter
4024: < 1. Here,
3516:. The confidence regions
3377:Let data be generated by
5831:. Chapman and Hall/CRC.
5772:The Annals of Statistics
5606:The Annals of Statistics
5176:Cox, D. R. (June 1958).
3373:Example 3: Binormal mean
16:Not to be confused with
6151:Lehmann, E.L. (1993). "
5232:10.1111/1467-9469.00285
5195:10.1214/aoms/1177706618
4928:10.1214/aoms/1177706618
4806:= (−∞,
4786:), 1 −
3631:{\displaystyle \gamma }
3611:{\displaystyle \gamma }
3560:{\displaystyle \gamma }
3509:{\displaystyle \gamma }
3422:{\displaystyle \gamma }
2277:correlation coefficient
1088:{\displaystyle t_{n-1}}
953:{\displaystyle \gamma }
620:{\displaystyle \gamma }
520:{\displaystyle \gamma }
29:confidence distribution
5828:Theoretical Statistics
5581:Cite journal requires
5406:Cite journal requires
5157:296–325. Reprinted in
4526:
4403:
4276:
4148:
4102:
4062:
4010: > 0 and
3972:
3860:
3763:
3659:
3632:
3612:
3592:
3561:
3537:
3510:
3490:
3447:
3423:
3403:
3359:
3327:
3287:
3038:
3000:
2978:
2634:
2448:
2353:
2253:
2139:
2036:
1998:
1946:
1813:
1773:
1729:
1675:
1655:
1604:
1553:
1517:
1470:
1366:
1163:
1127:
1089:
1056:
954:
934:
882:
851:
831:
804:
784:
764:
744:
700:
673:
641:
621:
601:
571:
521:
498:
66:functions, normalized
6226:Parametric statistics
6116:J. Roy. Statist. Soc.
5147:Phil. Trans. Roy. Soc
5104:Phil. Trans. Roy. Soc
4773:)} = 2 min{
4527:
4404:
4277:
4149:
4103:
4063:
3973:
3861:
3764:
3658:
3633:
3613:
3593:
3562:
3538:
3536:{\displaystyle A_{p}}
3511:
3491:
3448:
3424:
3404:
3360:
3328:
3288:
3039:
3001:
2979:
2635:
2449:
2364:limiting distribution
2354:
2289:Fisher transformation
2254:
2140:
2037:
1999:
1947:
1814:
1774:
1730:
1676:
1656:
1605:
1554:
1518:
1471:
1367:
1164:
1128:
1090:
1057:
955:
935:
883:
881:{\displaystyle A_{p}}
852:
832:
830:{\displaystyle A_{p}}
805:
785:
765:
745:
701:
699:{\displaystyle A_{p}}
674:
642:
622:
602:
600:{\displaystyle A_{p}}
572:
522:
499:
447:is an asymptotic CD (
230:The modern definition
37:fiducial distribution
25:statistical inference
6122:Hannig, J. (2009). "
6102:. New York: Hafner.
6098:Fisher, R A (1956).
5344:Annals of Statistics
4975:Statistical Science.
4878:Coverage probability
4445:
4289:
4201:
4116:
4076:
4028:
3882:
3768:
3666:
3622:
3602:
3575:
3551:
3520:
3500:
3461:
3437:
3413:
3381:
3337:
3299:
3057:
3010:
2990:
2660:
2472:
2462:is the sample size.
2369:
2298:
2153:
2053:
2008:
1963:
1837:
1787:
1747:
1701:
1674:{\displaystyle \mu }
1665:
1614:
1563:
1527:
1491:
1386:
1176:
1137:
1099:
1066:
1026:
944:
896:
865:
841:
814:
794:
774:
754:
710:
683:
651:
631:
611:
584:
539:
511:
488:
321:(R1) For each given
117:Classical definition
91:confidence intervals
68:likelihood functions
5141:Bayes, T. (1763). "
5120:Statistical Science
4741:. Thus, 2 min{
4386:
4243:
4048:
4003: > 0,
3985: −
3942:
3905:
3829:
3791:
3740:
3698:
3651:Confidence interval
3222:
2031:
1991:
1900:
1819:is an aCD for
531:is a distribution
443:Also, the function
93:, critical values,
52:confidence interval
27:, the concept of a
18:Confidence interval
5795:10.1214/13-AOS1083
5440:(540): 2086–2104.
5051:10.1111/insr.12000
4963:10.1111/insr.12001
4522:
4416:Hypothesis testing
4399:
4374:
4329:
4272:
4226:
4144:
4098:
4058:
4031:
3968:
3925:
3888:
3856:
3812:
3774:
3759:
3723:
3681:
3660:
3628:
3608:
3588:
3557:
3533:
3506:
3486:
3443:
3419:
3399:
3355:
3323:
3283:
3199:
3034:
2996:
2974:
2630:
2444:
2349:
2249:
2135:
2032:
2011:
1994:
1971:
1942:
1880:
1826:For the parameter
1809:
1779:is still a CD for
1769:
1725:
1693:For the parameter
1671:
1651:
1600:
1549:
1513:
1466:
1362:
1159:
1123:
1085:
1052:
950:
930:
878:
847:
827:
800:
780:
760:
740:
696:
669:
637:
617:
597:
579:confidence regions
567:
517:
494:
83:Bayesian inference
6221:Estimation theory
6128:Statistica Sinica
5846:978-0-429-17021-8
5629:10.1214/07-AOS501
5074:(sup1): 135–147.
4320:
4302:
4214:
4108:or it solves for
3446:{\displaystyle U}
3277:
3244:
3197:
3170:
3125:
2999:{\displaystyle F}
2969:
2945:
2926:
2910:
2893:
2849:
2804:
2763:
2757:
2735:
2618:
2586:
2573:
2541:
2526:
2439:
2418:
2386:
2347:
2315:
2243:
2233:
2213:
2129:
2119:
2099:
1632:
1581:
1460:
1450:
1430:
1353:
1343:
1323:
1263:
1250:
1240:
1220:
850:{\displaystyle C}
803:{\displaystyle C}
783:{\displaystyle I}
763:{\displaystyle C}
640:{\displaystyle p}
497:{\displaystyle C}
95:statistical power
6233:
6055:
6054:
6052:
6051:
6042:. Archived from
6040:www.krothman.org
6032:
6026:
6025:
6024:
6023:
6010:
6004:
6003:
6002:
6001:
5988:
5982:
5981:
5979:
5978:
5964:
5958:
5957:
5956:
5955:
5942:
5933:
5926:
5920:
5919:
5909:
5891:
5881:
5857:
5851:
5850:
5822:
5816:
5815:
5797:
5787:
5763:
5754:
5753:
5705:
5699:
5698:
5692:
5684:
5656:
5650:
5649:
5631:
5621:
5597:
5591:
5590:
5584:
5579:
5577:
5569:
5553:
5547:
5546:
5528:
5504:
5498:
5477:
5466:
5465:
5425:
5416:
5415:
5409:
5404:
5402:
5394:
5378:
5372:
5365:
5359:
5339:
5330:
5327:
5321:
5306:
5297:
5279:
5273:
5263:
5254:
5243:
5234:
5217:
5208:
5207:
5197:
5173:
5167:
5139:
5133:
5116:
5110:
5100:
5094:
5093:
5083:
5059:
5053:
5035:
5014:
4994:
4988:
4971:
4965:
4947:
4930:
4916:
4910:
4897:
4868:
4859:
4849:
4845:
4841:
4531:
4529:
4528:
4523:
4506:
4501:
4500:
4479:
4478:
4457:
4456:
4408:
4406:
4405:
4400:
4382:
4361:
4360:
4339:
4338:
4328:
4310:
4309:
4304:
4303:
4295:
4281:
4279:
4278:
4273:
4262:
4261:
4252:
4242:
4237:
4222:
4221:
4216:
4215:
4207:
4197:(1/2), the mean
4158:Point estimation
4153:
4151:
4150:
4145:
4128:
4127:
4107:
4105:
4104:
4099:
4088:
4087:
4067:
4065:
4064:
4059:
4047:
4039:
3977:
3975:
3974:
3971:{\displaystyle }
3969:
3961:
3960:
3941:
3933:
3918:
3917:
3904:
3896:
3865:
3863:
3862:
3859:{\displaystyle }
3857:
3846:
3828:
3820:
3802:
3790:
3782:
3766:
3765:
3760:
3739:
3731:
3697:
3689:
3637:
3635:
3634:
3629:
3617:
3615:
3614:
3609:
3597:
3595:
3594:
3589:
3587:
3586:
3566:
3564:
3563:
3558:
3542:
3540:
3539:
3534:
3532:
3531:
3515:
3513:
3512:
3507:
3495:
3493:
3492:
3487:
3473:
3472:
3452:
3450:
3449:
3444:
3428:
3426:
3425:
3420:
3408:
3406:
3405:
3400:
3364:
3362:
3361:
3356:
3332:
3330:
3329:
3324:
3292:
3290:
3289:
3284:
3282:
3278:
3276:
3269:
3268:
3258:
3245:
3237:
3228:
3221:
3210:
3198:
3196:
3173:
3172:
3171:
3166:
3155:
3149:
3148:
3127:
3126:
3121:
3110:
3104:
3103:
3084:
3073:
3043:
3041:
3040:
3035:
3005:
3003:
3002:
2997:
2983:
2981:
2980:
2975:
2970:
2965:
2951:
2946:
2938:
2927:
2919:
2911:
2903:
2895:
2894:
2889:
2875:
2851:
2850:
2845:
2834:
2828:
2827:
2806:
2805:
2800:
2789:
2783:
2782:
2764:
2762:
2758:
2750:
2736:
2728:
2725:
2687:
2676:
2639:
2637:
2636:
2631:
2629:
2625:
2624:
2620:
2619:
2617:
2606:
2595:
2587:
2579:
2574:
2572:
2561:
2550:
2542:
2534:
2527:
2516:
2509:
2508:
2484:
2483:
2453:
2451:
2450:
2445:
2440:
2438:
2424:
2419:
2417:
2406:
2395:
2387:
2379:
2358:
2356:
2355:
2350:
2348:
2346:
2335:
2324:
2316:
2308:
2281:bivariate normal
2258:
2256:
2255:
2250:
2248:
2244:
2239:
2235:
2234:
2226:
2214:
2209:
2206:
2200:
2199:
2198:
2197:
2165:
2164:
2144:
2142:
2141:
2136:
2134:
2130:
2125:
2121:
2120:
2112:
2100:
2095:
2092:
2086:
2085:
2067:
2066:
2065:
2041:
2039:
2038:
2033:
2030:
2025:
2003:
2001:
2000:
1995:
1993:
1992:
1990:
1985:
1951:
1949:
1948:
1943:
1935:
1930:
1929:
1902:
1901:
1899:
1894:
1856:
1855:
1854:
1853:
1818:
1816:
1815:
1810:
1799:
1798:
1778:
1776:
1775:
1770:
1759:
1758:
1734:
1732:
1731:
1726:
1715:
1714:
1713:
1681:, respectively.
1680:
1678:
1677:
1672:
1660:
1658:
1657:
1652:
1647:
1646:
1634:
1633:
1625:
1609:
1607:
1606:
1601:
1596:
1595:
1583:
1582:
1574:
1558:
1556:
1555:
1550:
1539:
1538:
1522:
1520:
1519:
1514:
1503:
1502:
1475:
1473:
1472:
1467:
1465:
1461:
1456:
1452:
1451:
1443:
1431:
1426:
1423:
1417:
1416:
1398:
1397:
1379:. Furthermore,
1371:
1369:
1368:
1363:
1358:
1354:
1349:
1345:
1344:
1336:
1324:
1319:
1316:
1310:
1309:
1308:
1307:
1275:
1274:
1264:
1261:
1255:
1251:
1246:
1242:
1241:
1233:
1221:
1216:
1213:
1207:
1206:
1188:
1187:
1168:
1166:
1165:
1160:
1149:
1148:
1132:
1130:
1129:
1124:
1113:
1112:
1111:
1094:
1092:
1091:
1086:
1084:
1083:
1061:
1059:
1058:
1053:
1051:
1050:
1049:
1048:
959:
957:
956:
951:
939:
937:
936:
931:
920:
919:
887:
885:
884:
879:
877:
876:
856:
854:
853:
848:
836:
834:
833:
828:
826:
825:
809:
807:
806:
801:
789:
787:
786:
781:
769:
767:
766:
761:
749:
747:
746:
741:
706:only exists for
705:
703:
702:
697:
695:
694:
678:
676:
675:
670:
646:
644:
643:
638:
626:
624:
623:
618:
606:
604:
603:
598:
596:
595:
577:for a family of
576:
574:
573:
568:
557:
556:
529:measurable space
526:
524:
523:
518:
503:
501:
500:
495:
464:(•) is dropped.
193:(•) =
6241:
6240:
6236:
6235:
6234:
6232:
6231:
6230:
6211:
6210:
6209:
6064:
6059:
6058:
6049:
6047:
6034:
6033:
6029:
6021:
6019:
6011:
6007:
5999:
5997:
5989:
5985:
5976:
5974:
5966:
5965:
5961:
5953:
5951:
5943:
5936:
5927:
5923:
5858:
5854:
5847:
5823:
5819:
5764:
5757:
5706:
5702:
5686:
5685:
5673:
5657:
5653:
5598:
5594:
5582:
5580:
5571:
5570:
5554:
5550:
5505:
5501:
5478:
5469:
5426:
5419:
5407:
5405:
5396:
5395:
5379:
5375:
5366:
5362:
5340:
5333:
5328:
5324:
5307:
5300:
5280:
5276:
5264:
5257:
5244:
5237:
5218:
5211:
5174:
5170:
5166:(1958) 293–315.
5140:
5136:
5117:
5113:
5101:
5097:
5060:
5056:
5036:
5017:
4995:
4991:
4972:
4968:
4948:
4933:
4917:
4913:
4898:
4891:
4886:
4874:
4866:
4857:
4847:
4844:pvaluefunctions
4843:
4839:
4830:
4828:Implementations
4816:
4805:
4794:
4781:
4772:
4765:
4756:
4749:
4733:)} ≤
4732:
4725:
4721:
4714:
4705:
4695:
4675:
4660:
4646:
4633:
4612:
4603:
4595:
4565:
4550:
4540:
4502:
4496:
4492:
4474:
4470:
4452:
4448:
4446:
4443:
4442:
4432:
4418:
4378:
4356:
4352:
4334:
4330:
4324:
4305:
4294:
4293:
4292:
4290:
4287:
4286:
4257:
4253:
4248:
4238:
4230:
4217:
4206:
4205:
4204:
4202:
4199:
4198:
4196:
4187:
4170:
4160:
4123:
4119:
4117:
4114:
4113:
4083:
4079:
4077:
4074:
4073:
4040:
4035:
4029:
4026:
4025:
4023:
4016:
4009:
4002:
3991:
3984:
3956:
3952:
3934:
3929:
3913:
3909:
3897:
3892:
3883:
3880:
3879:
3842:
3821:
3816:
3798:
3783:
3778:
3769:
3732:
3727:
3690:
3685:
3667:
3664:
3663:
3653:
3648:
3623:
3620:
3619:
3603:
3600:
3599:
3582:
3578:
3576:
3573:
3572:
3552:
3549:
3548:
3527:
3523:
3521:
3518:
3517:
3501:
3498:
3497:
3468:
3464:
3462:
3459:
3458:
3438:
3435:
3434:
3414:
3411:
3410:
3382:
3379:
3378:
3375:
3338:
3335:
3334:
3300:
3297:
3296:
3264:
3260:
3259:
3236:
3229:
3227:
3223:
3211:
3203:
3174:
3156:
3154:
3150:
3144:
3140:
3111:
3109:
3105:
3099:
3095:
3085:
3083:
3069:
3058:
3055:
3054:
3011:
3008:
3007:
2991:
2988:
2987:
2952:
2950:
2937:
2918:
2902:
2876:
2874:
2870:
2835:
2833:
2829:
2823:
2819:
2790:
2788:
2784:
2778:
2774:
2749:
2727:
2726:
2688:
2686:
2672:
2661:
2658:
2657:
2607:
2596:
2594:
2578:
2562:
2551:
2549:
2533:
2532:
2528:
2515:
2514:
2510:
2504:
2503:
2479:
2475:
2473:
2470:
2469:
2428:
2423:
2407:
2396:
2394:
2378:
2370:
2367:
2366:
2336:
2325:
2323:
2307:
2299:
2296:
2295:
2287:defined by the
2269:
2225:
2224:
2208:
2207:
2205:
2201:
2187:
2183:
2182:
2178:
2160:
2156:
2154:
2151:
2150:
2111:
2110:
2094:
2093:
2091:
2087:
2081:
2080:
2061:
2060:
2056:
2054:
2051:
2050:
2042:distribution .
2026:
2015:
2009:
2006:
2005:
1986:
1975:
1970:
1966:
1964:
1961:
1960:
1931:
1925:
1921:
1895:
1884:
1879:
1875:
1849:
1845:
1844:
1840:
1838:
1835:
1834:
1794:
1790:
1788:
1785:
1784:
1754:
1750:
1748:
1745:
1744:
1709:
1708:
1704:
1702:
1699:
1698:
1666:
1663:
1662:
1642:
1638:
1624:
1623:
1615:
1612:
1611:
1591:
1587:
1573:
1572:
1564:
1561:
1560:
1534:
1530:
1528:
1525:
1524:
1498:
1494:
1492:
1489:
1488:
1442:
1441:
1425:
1424:
1422:
1418:
1412:
1411:
1393:
1389:
1387:
1384:
1383:
1335:
1334:
1318:
1317:
1315:
1311:
1297:
1293:
1292:
1288:
1270:
1266:
1260:
1232:
1231:
1215:
1214:
1212:
1208:
1202:
1201:
1183:
1179:
1177:
1174:
1173:
1144:
1140:
1138:
1135:
1134:
1107:
1106:
1102:
1100:
1097:
1096:
1073:
1069:
1067:
1064:
1063:
1038:
1034:
1033:
1029:
1027:
1024:
1023:
985:
971:
966:
945:
942:
941:
915:
911:
897:
894:
893:
872:
868:
866:
863:
862:
842:
839:
838:
821:
817:
815:
812:
811:
795:
792:
791:
775:
772:
771:
755:
752:
751:
711:
708:
707:
690:
686:
684:
681:
680:
652:
649:
648:
647:for all levels
632:
629:
628:
612:
609:
608:
591:
587:
585:
582:
581:
552:
548:
540:
537:
536:
512:
509:
508:
489:
486:
485:
482:
463:
432:
421:
414:
403:
394:
387:
378:
360:
351:
342:
329:
305:
294:
285:
272:
263:
256:
232:
201:
192:
183:
174:
167:
154:
137:
119:
114:
103:
87:point estimates
48:point estimator
21:
12:
11:
5:
6239:
6229:
6228:
6223:
6208:
6207:
6200:
6189:
6178:
6163:
6149:
6135:
6120:
6111:
6096:
6080:
6065:
6063:
6060:
6057:
6056:
6027:
6005:
5983:
5959:
5934:
5921:
5852:
5845:
5837:10.1201/b14832
5817:
5755:
5720:(2): 170–193.
5700:
5671:
5651:
5592:
5583:|journal=
5548:
5499:
5467:
5417:
5408:|journal=
5373:
5360:
5331:
5322:
5298:
5274:
5255:
5235:
5209:
5188:(2): 357–372.
5168:
5134:
5111:
5095:
5054:
5015:
4989:
4966:
4931:
4911:
4888:
4887:
4885:
4882:
4881:
4880:
4873:
4870:
4829:
4826:
4817: = [
4814:
4803:
4790:
4777:
4770:
4761:
4754:
4745:
4737:) =
4730:
4723:
4719:
4710:
4693:
4688:
4673:
4658:
4642:
4638:) =
4629:
4621:) =
4617:) ≤
4608:
4599:
4587:
4563:
4548:
4536:
4521:
4518:
4515:
4512:
4509:
4505:
4499:
4495:
4491:
4488:
4485:
4482:
4477:
4473:
4469:
4466:
4463:
4460:
4455:
4451:
4428:
4417:
4414:
4410:
4409:
4398:
4395:
4392:
4389:
4385:
4381:
4377:
4373:
4370:
4367:
4364:
4359:
4355:
4351:
4348:
4345:
4342:
4337:
4333:
4327:
4323:
4319:
4316:
4313:
4308:
4301:
4298:
4271:
4268:
4265:
4260:
4256:
4251:
4246:
4241:
4236:
4233:
4229:
4225:
4220:
4213:
4210:
4192:
4183:
4166:
4159:
4156:
4143:
4140:
4137:
4134:
4131:
4126:
4122:
4097:
4094:
4091:
4086:
4082:
4072:% quantile of
4057:
4054:
4051:
4046:
4043:
4038:
4034:
4021:
4014:
4007:
4000:
3989:
3982:
3967:
3964:
3959:
3955:
3951:
3948:
3945:
3940:
3937:
3932:
3928:
3924:
3921:
3916:
3912:
3908:
3903:
3900:
3895:
3891:
3887:
3855:
3852:
3849:
3845:
3841:
3838:
3835:
3832:
3827:
3824:
3819:
3815:
3811:
3808:
3805:
3801:
3797:
3794:
3789:
3786:
3781:
3777:
3773:
3758:
3755:
3752:
3749:
3746:
3743:
3738:
3735:
3730:
3726:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3696:
3693:
3688:
3684:
3680:
3677:
3674:
3671:
3652:
3649:
3647:
3644:
3627:
3607:
3585:
3581:
3556:
3530:
3526:
3505:
3485:
3482:
3479:
3476:
3471:
3467:
3442:
3418:
3398:
3395:
3392:
3389:
3386:
3374:
3371:
3354:
3351:
3348:
3345:
3342:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3281:
3275:
3272:
3267:
3263:
3257:
3254:
3251:
3248:
3243:
3240:
3235:
3232:
3226:
3220:
3217:
3214:
3209:
3206:
3202:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3169:
3165:
3162:
3159:
3153:
3147:
3143:
3139:
3136:
3133:
3130:
3124:
3120:
3117:
3114:
3108:
3102:
3098:
3094:
3091:
3088:
3082:
3079:
3076:
3072:
3068:
3065:
3062:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
2995:
2973:
2968:
2964:
2961:
2958:
2955:
2949:
2944:
2941:
2936:
2933:
2930:
2925:
2922:
2917:
2914:
2909:
2906:
2901:
2898:
2892:
2888:
2885:
2882:
2879:
2873:
2869:
2866:
2863:
2860:
2857:
2854:
2848:
2844:
2841:
2838:
2832:
2826:
2822:
2818:
2815:
2812:
2809:
2803:
2799:
2796:
2793:
2787:
2781:
2777:
2773:
2770:
2767:
2761:
2756:
2753:
2748:
2745:
2742:
2739:
2734:
2731:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2685:
2682:
2679:
2675:
2671:
2668:
2665:
2641:
2640:
2628:
2623:
2616:
2613:
2610:
2605:
2602:
2599:
2593:
2590:
2585:
2582:
2577:
2571:
2568:
2565:
2560:
2557:
2554:
2548:
2545:
2540:
2537:
2531:
2525:
2522:
2519:
2513:
2507:
2502:
2499:
2496:
2493:
2490:
2487:
2482:
2478:
2443:
2437:
2434:
2431:
2427:
2422:
2416:
2413:
2410:
2405:
2402:
2399:
2393:
2390:
2385:
2382:
2377:
2374:
2360:
2359:
2345:
2342:
2339:
2334:
2331:
2328:
2322:
2319:
2314:
2311:
2306:
2303:
2268:
2265:
2247:
2242:
2238:
2232:
2229:
2223:
2220:
2217:
2212:
2204:
2196:
2193:
2190:
2186:
2181:
2177:
2174:
2171:
2168:
2163:
2159:
2133:
2128:
2124:
2118:
2115:
2109:
2106:
2103:
2098:
2090:
2084:
2079:
2076:
2073:
2070:
2064:
2059:
2029:
2024:
2021:
2018:
2014:
1989:
1984:
1981:
1978:
1974:
1969:
1953:
1952:
1941:
1938:
1934:
1928:
1924:
1920:
1917:
1914:
1911:
1908:
1905:
1898:
1893:
1890:
1887:
1883:
1878:
1874:
1871:
1868:
1865:
1862:
1859:
1852:
1848:
1843:
1808:
1805:
1802:
1797:
1793:
1768:
1765:
1762:
1757:
1753:
1724:
1721:
1718:
1712:
1707:
1670:
1650:
1645:
1641:
1637:
1631:
1628:
1622:
1619:
1599:
1594:
1590:
1586:
1580:
1577:
1571:
1568:
1548:
1545:
1542:
1537:
1533:
1512:
1509:
1506:
1501:
1497:
1487:. The uses of
1477:
1476:
1464:
1459:
1455:
1449:
1446:
1440:
1437:
1434:
1429:
1421:
1415:
1410:
1407:
1404:
1401:
1396:
1392:
1373:
1372:
1361:
1357:
1352:
1348:
1342:
1339:
1333:
1330:
1327:
1322:
1314:
1306:
1303:
1300:
1296:
1291:
1287:
1284:
1281:
1278:
1273:
1269:
1258:
1254:
1249:
1245:
1239:
1236:
1230:
1227:
1224:
1219:
1211:
1205:
1200:
1197:
1194:
1191:
1186:
1182:
1158:
1155:
1152:
1147:
1143:
1122:
1119:
1116:
1110:
1105:
1082:
1079:
1076:
1072:
1047:
1044:
1041:
1037:
1032:
981:
970:
967:
965:
962:
949:
929:
926:
923:
918:
914:
910:
907:
904:
901:
875:
871:
859:random measure
846:
824:
820:
799:
779:
759:
739:
736:
733:
730:
727:
724:
721:
718:
715:
693:
689:
668:
665:
662:
659:
656:
636:
616:
594:
590:
566:
563:
560:
555:
551:
547:
544:
516:
493:
481:
478:
459:
441:
440:
439:
438:
428:
419:
410:
399:
395:) ≡
392:
383:
376:
366:
356:
347:
338:
325:
301:
290:
281:
268:
261:
252:
231:
228:
208:
207:
197:
188:
179:
172:
163:
150:
133:
118:
115:
113:
110:
102:
99:
9:
6:
4:
3:
2:
6238:
6227:
6224:
6222:
6219:
6218:
6216:
6205:
6201:
6198:
6194:
6190:
6188:2010 15: 1–19
6187:
6183:
6179:
6176:
6172:
6168:
6164:
6161:
6158:
6154:
6150:
6147:
6144:
6140:
6136:
6133:
6129:
6125:
6121:
6118:
6117:
6112:
6109:
6108:0-02-844740-9
6105:
6101:
6097:
6095:
6094:9781139046671
6091:
6087:
6084:
6081:
6078:
6074:
6070:
6067:
6066:
6046:on 2020-01-29
6045:
6041:
6037:
6031:
6018:
6017:
6009:
5996:
5995:
5987:
5973:
5969:
5963:
5950:
5949:
5941:
5939:
5931:
5925:
5917:
5913:
5908:
5903:
5899:
5895:
5890:
5885:
5880:
5875:
5871:
5867:
5863:
5856:
5848:
5842:
5838:
5834:
5830:
5829:
5821:
5813:
5809:
5805:
5801:
5796:
5791:
5786:
5781:
5777:
5773:
5769:
5762:
5760:
5751:
5747:
5743:
5739:
5735:
5731:
5727:
5723:
5719:
5715:
5711:
5704:
5696:
5690:
5682:
5678:
5674:
5672:0-02-844740-9
5668:
5664:
5663:
5655:
5647:
5643:
5639:
5635:
5630:
5625:
5620:
5615:
5611:
5607:
5603:
5596:
5588:
5575:
5567:
5563:
5559:
5552:
5544:
5540:
5536:
5532:
5527:
5522:
5518:
5514:
5510:
5503:
5497:
5493:
5489:
5485:
5482:
5476:
5474:
5472:
5463:
5459:
5455:
5451:
5447:
5443:
5439:
5435:
5431:
5424:
5422:
5413:
5400:
5392:
5388:
5384:
5377:
5370:
5364:
5358:
5354:
5350:
5346:
5345:
5338:
5336:
5326:
5320:
5318:
5313:
5310:
5305:
5303:
5295:
5294:0-521-68567-2
5291:
5287:
5283:
5278:
5272:
5268:
5262:
5260:
5252:
5248:
5242:
5240:
5233:
5229:
5225:
5222:
5216:
5214:
5205:
5201:
5196:
5191:
5187:
5183:
5179:
5172:
5165:
5162:
5161:
5156:
5152:
5148:
5144:
5138:
5132:
5128:
5124:
5121:
5115:
5108:
5105:
5099:
5091:
5087:
5082:
5077:
5073:
5069:
5065:
5058:
5052:
5048:
5044:
5040:
5034:
5032:
5030:
5028:
5026:
5024:
5022:
5020:
5013:
5009:
5005:
5001:
5000:
4993:
4987:
4983:
4979:
4976:
4970:
4964:
4960:
4956:
4952:
4946:
4944:
4942:
4940:
4938:
4936:
4929:
4925:
4921:
4915:
4908:
4905:
4901:
4896:
4894:
4889:
4879:
4876:
4875:
4869:
4864:
4860:
4855:
4851:
4837:
4833:
4825:
4822:
4820:
4813:
4809:
4802:
4798:
4793:
4789:
4785:
4780:
4776:
4769:
4764:
4760:
4753:
4748:
4744:
4740:
4736:
4729:
4718:
4713:
4709:
4703:
4700: =
4699:
4692:
4687:
4683:
4680: ≠
4679:
4672:
4668:
4665: =
4664:
4657:
4652:
4650:
4645:
4641:
4637:
4632:
4628:
4624:
4620:
4616:
4611:
4607:
4602:
4598:
4594:
4591: ∈
4590:
4585:
4581:
4577:
4573:
4570: ∈
4569:
4562:
4558:
4555: ∈
4554:
4547:
4542:
4539:
4535:
4519:
4513:
4507:
4497:
4493:
4489:
4483:
4475:
4471:
4467:
4461:
4453:
4449:
4440:
4436:
4431:
4427:
4423:
4413:
4396:
4390:
4383:
4379:
4375:
4371:
4365:
4357:
4353:
4349:
4343:
4335:
4331:
4325:
4317:
4314:
4311:
4306:
4299:
4296:
4285:
4284:
4283:
4266:
4258:
4254:
4244:
4231:
4227:
4223:
4218:
4208:
4195:
4191:
4188: =
4186:
4182:
4178:
4174:
4169:
4165:
4155:
4141:
4138:
4132:
4124:
4120:
4111:
4092:
4084:
4080:
4071:
4052:
4044:
4041:
4036:
4032:
4020:
4017: +
4013:
4006:
3999:
3995:
3988:
3981:
3957:
3953:
3949:
3946:
3938:
3935:
3930:
3926:
3922:
3914:
3910:
3901:
3898:
3893:
3889:
3877:
3873:
3869:
3847:
3843:
3839:
3836:
3833:
3825:
3822:
3817:
3813:
3809:
3803:
3799:
3795:
3787:
3784:
3779:
3775:
3750:
3744:
3736:
3733:
3728:
3724:
3717:
3708:
3705:
3702:
3694:
3691:
3686:
3682:
3678:
3672:
3657:
3643:
3641:
3640:Hilbert space
3625:
3605:
3583:
3570:
3554:
3546:
3528:
3524:
3503:
3483:
3480:
3477:
3474:
3469:
3456:
3440:
3432:
3416:
3396:
3393:
3390:
3387:
3384:
3370:
3368:
3352:
3349:
3346:
3343:
3340:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3293:
3279:
3273:
3270:
3265:
3261:
3255:
3252:
3249:
3246:
3241:
3238:
3233:
3230:
3224:
3218:
3215:
3212:
3207:
3204:
3193:
3187:
3184:
3181:
3175:
3167:
3163:
3160:
3157:
3145:
3141:
3137:
3134:
3128:
3122:
3118:
3115:
3112:
3100:
3096:
3092:
3089:
3080:
3074:
3066:
3060:
3052:
3050:
3045:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
2993:
2984:
2966:
2962:
2959:
2956:
2953:
2947:
2942:
2939:
2934:
2931:
2928:
2923:
2920:
2915:
2912:
2907:
2904:
2896:
2890:
2886:
2883:
2880:
2877:
2867:
2864:
2861:
2858:
2852:
2846:
2842:
2839:
2836:
2824:
2820:
2816:
2813:
2807:
2801:
2797:
2794:
2791:
2779:
2775:
2771:
2768:
2754:
2751:
2746:
2743:
2732:
2729:
2719:
2716:
2713:
2701:
2698:
2695:
2689:
2683:
2677:
2669:
2663:
2655:
2653:
2648:
2646:
2626:
2621:
2614:
2611:
2608:
2603:
2600:
2597:
2591:
2588:
2583:
2580:
2575:
2569:
2566:
2563:
2558:
2555:
2552:
2546:
2543:
2538:
2535:
2529:
2523:
2520:
2517:
2511:
2500:
2497:
2494:
2488:
2480:
2476:
2468:
2467:
2466:
2465:The function
2463:
2461:
2457:
2435:
2432:
2429:
2425:
2420:
2414:
2411:
2408:
2403:
2400:
2397:
2391:
2388:
2383:
2380:
2372:
2365:
2343:
2340:
2337:
2332:
2329:
2326:
2320:
2317:
2312:
2309:
2304:
2301:
2294:
2293:
2292:
2290:
2286:
2282:
2278:
2274:
2264:
2262:
2245:
2240:
2227:
2221:
2218:
2210:
2202:
2194:
2191:
2188:
2184:
2179:
2175:
2169:
2161:
2157:
2148:
2131:
2126:
2113:
2107:
2104:
2096:
2088:
2077:
2071:
2057:
2048:
2043:
2027:
2022:
2019:
2016:
2012:
1987:
1982:
1979:
1976:
1972:
1967:
1958:
1936:
1932:
1926:
1922:
1915:
1912:
1909:
1896:
1891:
1888:
1885:
1881:
1876:
1872:
1869:
1866:
1860:
1850:
1846:
1841:
1833:
1832:
1831:
1829:
1824:
1822:
1803:
1795:
1791:
1782:
1763:
1755:
1751:
1742:
1738:
1719:
1705:
1696:
1691:
1690:
1688:
1685:(2) Variance
1682:
1668:
1643:
1639:
1635:
1626:
1617:
1592:
1588:
1584:
1575:
1566:
1543:
1535:
1531:
1507:
1499:
1495:
1486:
1482:
1462:
1457:
1444:
1438:
1435:
1427:
1419:
1408:
1402:
1394:
1390:
1382:
1381:
1380:
1378:
1359:
1355:
1350:
1337:
1331:
1328:
1320:
1312:
1304:
1301:
1298:
1294:
1289:
1285:
1279:
1271:
1267:
1256:
1252:
1247:
1234:
1228:
1225:
1217:
1209:
1198:
1192:
1180:
1172:
1171:
1170:
1153:
1145:
1141:
1117:
1103:
1080:
1077:
1074:
1070:
1045:
1042:
1039:
1035:
1030:
1021:
1016:
1015:
1013:
1010:(1) Variance
1007:
1005:
1001:
997:
993:
989:
986: ~
984:
980:
976:
961:
947:
927:
924:
916:
912:
908:
905:
899:
891:
873:
869:
860:
844:
822:
818:
797:
777:
757:
734:
731:
728:
722:
719:
716:
713:
691:
687:
666:
663:
660:
657:
654:
634:
614:
592:
588:
580:
564:
561:
553:
549:
542:
534:
530:
514:
507:
491:
477:
473:
469:
465:
462:
458:
454:
450:
446:
436:
431:
427:
426:
418:
413:
409:
408:
402:
398:
391:
386:
382:
375:
372: =
371:
367:
364:
359:
355:
350:
346:
341:
337:
333:
328:
324:
320:
319:
317:
313:
310: ×
309:
306:, •) on
304:
300:
299:
293:
289:
284:
280:
276:
275:
274:
271:
267:
260:
255:
251:
250:
245:
241:
237:
227:
223:
221:
217:
213:
205:
200:
196:
191:
187:
182:
178:
171:
166:
162:
158:
153:
149:
145:
141:
136:
132:
128:
124:
123:
122:
109:
107:
98:
96:
92:
88:
84:
79:
77:
74:and Bayesian
73:
69:
65:
60:
55:
53:
49:
45:
40:
38:
34:
30:
26:
19:
6196:
6185:
6166:
6159:
6156:
6145:
6142:
6131:
6127:
6114:
6099:
6085:
6076:
6072:
6062:Bibliography
6048:. Retrieved
6044:the original
6039:
6030:
6020:, retrieved
6015:
6008:
5998:, retrieved
5993:
5986:
5975:. Retrieved
5971:
5962:
5952:, retrieved
5947:
5929:
5924:
5869:
5865:
5855:
5827:
5820:
5775:
5771:
5717:
5713:
5703:
5661:
5654:
5609:
5605:
5595:
5574:cite journal
5551:
5516:
5512:
5502:
5487:
5483:
5480:
5437:
5433:
5399:cite journal
5376:
5363:
5348:
5342:
5325:
5316:
5314:
5309:
5285:
5277:
5250:
5246:
5223:
5220:
5185:
5181:
5171:
5163:
5158:
5154:
5150:
5146:
5137:
5122:
5119:
5114:
5106:
5103:
5098:
5071:
5067:
5057:
5042:
5038:
5003:
4997:
4992:
4977:
4974:
4969:
4954:
4950:
4914:
4906:
4903:
4900:Fisher, R.A.
4861:
4852:
4834:
4831:
4823:
4818:
4811:
4807:
4800:
4796:
4791:
4787:
4783:
4778:
4774:
4767:
4762:
4758:
4751:
4746:
4742:
4738:
4734:
4727:
4716:
4711:
4707:
4706:(2 min{
4701:
4697:
4690:
4685:
4681:
4677:
4670:
4666:
4662:
4655:
4653:
4648:
4643:
4639:
4635:
4630:
4626:
4622:
4618:
4614:
4609:
4605:
4600:
4596:
4592:
4588:
4583:
4579:
4575:
4571:
4567:
4560:
4556:
4552:
4545:
4543:
4537:
4533:
4438:
4434:
4429:
4425:
4421:
4419:
4411:
4193:
4189:
4184:
4180:
4176:
4172:
4167:
4163:
4161:
4112:in equation
4109:
4069:
4018:
4011:
4004:
3997:
3993:
3986:
3979:
3875:
3871:
3867:
3661:
3547:centered at
3376:
3294:
3053:
3046:
2985:
2656:
2651:
2649:
2644:
2642:
2464:
2459:
2455:
2361:
2284:
2275:denotes the
2272:
2270:
2260:
2149:is unknown,
2146:
2046:
2044:
1956:
1954:
1827:
1825:
1820:
1780:
1740:
1736:
1694:
1692:
1686:
1684:
1683:
1661:to estimate
1484:
1480:
1478:
1376:
1374:
1019:
1017:
1011:
1009:
1008:
1003:
999:
995:
991:
987:
982:
978:
972:
483:
474:
470:
466:
460:
456:
452:
448:
444:
442:
434:
429:
424:
423:
416:
411:
406:
405:
400:
396:
389:
384:
380:
373:
369:
362:
357:
353:
348:
344:
339:
335:
331:
326:
322:
315:
311:
307:
302:
297:
296:
291:
287:
282:
278:
269:
265:
258:
253:
248:
247:
243:
239:
235:
233:
224:
219:
215:
211:
209:
203:
198:
194:
189:
185:
180:
176:
169:
164:
160:
156:
151:
147:
143:
139:
134:
130:
126:
120:
104:
80:
56:
41:
32:
28:
22:
6143:Biometrika.
5519:: 600–616.
5351:, 159–183.
5296:. (page 66)
5125:, 299-316.
5006:, 258–265.
4821:, ∞).
1743:. However,
890:random set.
627:with level
277:A function
44:frequentist
6215:Categories
6199:, 346-353.
6162:1242–1249.
6134:, 491–544.
6050:2020-04-15
6022:2020-04-15
6000:2020-04-15
5977:2020-04-15
5954:2020-05-05
5879:1909.08579
5872:(1): 244.
5312:Biometrika
5282:Cox, D. R.
5247:Stat. Sci.
5160:Biometrika
4909:, 528–535.
4884:References
4838:, via the
4068:is the 100
3874:, for any
3571:matrix of
3569:covariance
2049:is known,
1689:is unknown
1006:is given.
973:Suppose a
451:), if the
125:For every
112:Definition
76:posteriors
5898:1471-2288
5804:0090-5364
5785:1301.1717
5750:120705955
5734:0976-836X
5689:cite book
5638:0090-5364
5619:0804.0987
5543:244594067
5535:0976-8378
5513:Sankhya A
5462:233657455
5454:0162-1459
5253:, 369–387
5226:309–332.
5204:0003-4851
5149:, London
5090:0003-1305
4957:, 68-77.
4850:packages
4514:θ
4494:∫
4391:θ
4366:θ
4344:θ
4326:θ
4318:
4300:^
4297:θ
4240:∞
4235:∞
4232:−
4228:∫
4212:¯
4209:θ
4142:β
4133:θ
4093:θ
4053:β
4042:−
3954:α
3950:−
3936:−
3911:α
3899:−
3840:α
3837:−
3823:−
3796:α
3785:−
3754:∞
3745:α
3734:−
3709:α
3706:−
3692:−
3676:∞
3673:−
3626:γ
3606:γ
3580:Γ
3555:γ
3504:γ
3481:−
3466:Γ
3417:γ
3391:γ
3367:C. R. Rao
3353:π
3347:θ
3318:ρ
3315:−
3309:θ
3306:
3274:θ
3271:
3256:θ
3250:
3234:−
3231:θ
3216:−
3213:ν
3205:ρ
3201:∂
3185:−
3182:ν
3176:π
3161:−
3158:ν
3142:ρ
3138:−
3129:⋅
3116:−
3113:ν
3093:−
3067:ρ
3061:π
3023:−
3014:ν
2963:ρ
2932:ν
2916:−
2887:ν
2881:−
2868:ρ
2862:−
2853:⋅
2840:−
2837:ν
2821:ρ
2817:−
2808:⋅
2795:−
2792:ν
2772:−
2744:ν
2738:Γ
2733:π
2717:−
2714:ν
2708:Γ
2699:−
2696:ν
2690:ν
2670:ρ
2664:π
2615:ρ
2612:−
2604:ρ
2592:
2576:−
2567:−
2547:
2521:−
2506:Φ
2501:−
2489:ρ
2433:−
2415:ρ
2412:−
2404:ρ
2392:
2341:−
2321:
2231:¯
2222:−
2219:μ
2192:−
2170:μ
2127:σ
2117:¯
2108:−
2105:μ
2083:Φ
2072:μ
2063:Φ
2020:−
2013:χ
1980:−
1973:χ
1937:θ
1913:−
1889:−
1882:χ
1873:−
1861:θ
1847:χ
1804:μ
1764:μ
1720:μ
1711:Φ
1669:μ
1630:¯
1589:σ
1579:¯
1544:μ
1508:μ
1448:¯
1439:−
1436:μ
1414:Φ
1403:μ
1341:¯
1332:−
1329:μ
1302:−
1280:μ
1248:σ
1238:¯
1229:−
1226:μ
1204:Φ
1193:μ
1185:Φ
1169:given by
1154:μ
1118:μ
1109:Φ
1078:−
1043:−
948:γ
925:≥
909:∈
906:γ
723:⊂
717:∈
615:γ
533:estimator
515:γ
506:parameter
159:) =
59:bootstrap
6148:529–542.
5916:32998683
5812:88520957
5742:42003718
5646:14703802
5496:20461464
5284:(2006).
5271:23059131
5153:370–418
5131:23059129
5045:, 3-39.
4980:95–122.
4872:See also
4867:concurve
4858:episheet
4848:episheet
4840:concurve
4757:),
4625:. Thus,
4574:, where
4384:′
3996:for any
3545:ellipses
3455:binormal
2362:has the
1959:. Here,
1697:, since
1014:is known
964:Examples
810:and all
184:. Then,
146:, where
6175:2983716
6079:, 3–39.
5907:7528258
5357:3448660
5288:, CUP.
5109:333–380
5012:2290557
4986:2290557
2986:where
994:,
977:sample
790:. Both
750:, then
264:, ...,
101:History
64:p-value
6173:
6106:
6092:
5914:
5904:
5896:
5843:
5810:
5802:
5748:
5740:
5732:
5681:785822
5679:
5669:
5644:
5636:
5541:
5533:
5494:
5460:
5452:
5355:
5292:
5269:
5202:
5129:
5088:
5010:
4984:
4865:, via
4856:, via
4846:, and
4810:] and
4582:] or [
3453:has a
3409:where
3295:where
3051:gives
3049:Fisher
975:normal
504:for a
343:(•) =
286:(•) =
242:, and
106:Neyman
72:priors
6171:JSTOR
6146:92(3)
5874:arXiv
5808:S2CID
5780:arXiv
5778:(1).
5746:S2CID
5738:JSTOR
5642:S2CID
5614:arXiv
5612:(2).
5539:S2CID
5492:JSTOR
5458:S2CID
5353:JSTOR
5319:3–26.
5267:JSTOR
5127:JSTOR
5008:JSTOR
4982:JSTOR
4863:Stata
4854:Excel
4532:This
3431:plane
2279:of a
1018:Let,
888:is a
857:is a
535:with
527:in a
6197:1305
6104:ISBN
6090:ISBN
5912:PMID
5894:ISSN
5841:ISBN
5800:ISSN
5730:ISSN
5695:link
5677:OCLC
5667:ISBN
5634:ISSN
5587:help
5531:ISSN
5450:ISSN
5412:help
5290:ISBN
5200:ISSN
5107:A237
5086:ISSN
4669:vs.
4559:vs.
3433:and
3350:<
3344:<
3333:and
3029:>
2271:Let
1783:and
1610:and
1523:and
1133:and
861:and
664:<
658:<
607:for
6141:."
6126:".
5902:PMC
5884:doi
5833:doi
5790:doi
5722:doi
5624:doi
5562:doi
5521:doi
5442:doi
5438:117
5387:doi
5228:doi
5190:doi
5145:."
5076:doi
5047:doi
4959:doi
4924:doi
4322:max
4315:arg
3303:cos
3262:sin
3247:sin
2654:is
1262:and
998:),
449:aCD
273:}:
216:etc
23:In
6217::
6184:"
6160:88
6155:"
6132:19
6130:,
6077:81
6075:,
6038:.
5970:.
5937:^
5910:.
5900:.
5892:.
5882:.
5870:20
5868:.
5864:.
5839:.
5806:.
5798:.
5788:.
5776:41
5774:.
5770:.
5758:^
5744:.
5736:.
5728:.
5718:74
5716:.
5712:.
5691:}}
5687:{{
5675:.
5640:.
5632:.
5622:.
5610:36
5608:.
5604:.
5578::
5576:}}
5572:{{
5560:.
5537:.
5529:.
5517:85
5515:.
5511:.
5488:54
5486:,
5470:^
5456:.
5448:.
5436:.
5432:.
5420:^
5403::
5401:}}
5397:{{
5385:.
5349:33
5347:,
5334:^
5317:80
5315:,
5301:^
5258:^
5249:,
5238:^
5224:29
5212:^
5198:.
5186:29
5184:.
5180:.
5164:45
5155:54
5151:53
5123:26
5084:.
5072:73
5070:.
5066:.
5043:81
5041:,
5018:^
5004:86
5002:,
4978:13
4955:81
4953:,
4934:^
4907:26
4892:^
4842:,
4815:up
4804:lo
4771:up
4755:lo
4731:up
4720:lo
4696::
4684:,
4676::
4661::
4566::
4551::
3369:.
2647:.
2589:ln
2544:ln
2389:ln
2318:ln
2291::
2263:.
1823:.
415:,
379:,
334:,
330:∈
257:={
89:,
78:.
33:CD
6110:.
6053:.
5980:.
5918:.
5886::
5876::
5849:.
5835::
5814:.
5792::
5782::
5752:.
5724::
5697:)
5683:.
5648:.
5626::
5616::
5589:)
5585:(
5568:.
5564::
5545:.
5523::
5464:.
5444::
5414:)
5410:(
5393:.
5389::
5251:7
5230::
5206:.
5192::
5092:.
5078::
5049::
4961::
4926::
4836:R
4819:b
4812:C
4808:b
4801:C
4797:b
4795:(
4792:n
4788:H
4784:b
4782:(
4779:n
4775:H
4768:C
4766:(
4763:s
4759:p
4752:C
4750:(
4747:s
4743:p
4739:α
4735:α
4728:C
4726:(
4724:s
4717:C
4715:(
4712:s
4708:p
4704:}
4702:b
4698:θ
4694:0
4691:K
4689:{
4686:P
4682:b
4678:θ
4674:1
4671:K
4667:b
4663:θ
4659:0
4656:K
4649:C
4647:(
4644:n
4640:H
4636:C
4634:(
4631:s
4627:p
4623:α
4619:α
4615:C
4613:(
4610:s
4606:p
4604:(
4601:θ
4597:P
4593:C
4589:θ
4584:b
4580:b
4576:C
4572:C
4568:θ
4564:1
4561:K
4557:C
4553:θ
4549:0
4546:K
4538:s
4534:p
4520:.
4517:)
4511:(
4508:H
4504:d
4498:C
4490:=
4487:)
4484:C
4481:(
4476:n
4472:H
4468:=
4465:)
4462:C
4459:(
4454:s
4450:p
4439:C
4435:θ
4433:(
4430:n
4426:H
4422:θ
4397:.
4394:)
4388:(
4380:n
4376:H
4372:=
4369:)
4363:(
4358:n
4354:h
4350:,
4347:)
4341:(
4336:n
4332:h
4312:=
4307:n
4270:)
4267:t
4264:(
4259:n
4255:H
4250:d
4245:t
4224:=
4219:n
4194:n
4190:H
4185:n
4181:M
4177:θ
4173:θ
4171:(
4168:n
4164:H
4139:=
4136:)
4130:(
4125:n
4121:H
4110:θ
4096:)
4090:(
4085:n
4081:H
4070:β
4056:)
4050:(
4045:1
4037:n
4033:H
4022:2
4019:α
4015:1
4012:α
4008:2
4005:α
4001:1
3998:α
3994:θ
3990:2
3987:α
3983:1
3980:α
3966:]
3963:)
3958:2
3947:1
3944:(
3939:1
3931:n
3927:H
3923:,
3920:)
3915:1
3907:(
3902:1
3894:n
3890:H
3886:[
3876:α
3872:θ
3868:α
3854:]
3851:)
3848:2
3844:/
3834:1
3831:(
3826:1
3818:n
3814:H
3810:,
3807:)
3804:2
3800:/
3793:(
3788:1
3780:n
3776:H
3772:[
3757:)
3751:,
3748:)
3742:(
3737:1
3729:n
3725:H
3721:[
3718:,
3715:]
3712:)
3703:1
3700:(
3695:1
3687:n
3683:H
3679:,
3670:(
3584:y
3529:p
3525:A
3484:U
3478:y
3475:=
3470:y
3441:U
3397:U
3394:+
3388:=
3385:Y
3341:0
3321:r
3312:=
3280:}
3266:3
3253:2
3242:2
3239:1
3225:{
3219:2
3208:r
3194:!
3191:)
3188:2
3179:(
3168:2
3164:2
3152:)
3146:2
3135:1
3132:(
3123:2
3119:1
3107:)
3101:2
3097:r
3090:1
3087:(
3081:=
3078:)
3075:r
3071:|
3064:(
3032:1
3026:1
3020:n
3017:=
2994:F
2972:)
2967:2
2960:r
2957:+
2954:1
2948:;
2943:2
2940:1
2935:+
2929:;
2924:2
2921:1
2913:,
2908:2
2905:3
2900:(
2897:F
2891:2
2884:2
2878:1
2872:)
2865:r
2859:1
2856:(
2847:2
2843:2
2831:)
2825:2
2814:1
2811:(
2802:2
2798:1
2786:)
2780:2
2776:r
2769:1
2766:(
2760:)
2755:2
2752:1
2747:+
2741:(
2730:2
2723:)
2720:1
2711:(
2705:)
2702:1
2693:(
2684:=
2681:)
2678:r
2674:|
2667:(
2652:ρ
2645:ρ
2627:)
2622:)
2609:1
2601:+
2598:1
2584:2
2581:1
2570:r
2564:1
2559:r
2556:+
2553:1
2539:2
2536:1
2530:(
2524:3
2518:n
2512:(
2498:1
2495:=
2492:)
2486:(
2481:n
2477:H
2460:n
2456:r
2442:)
2436:3
2430:n
2426:1
2421:,
2409:1
2401:+
2398:1
2384:2
2381:1
2376:(
2373:N
2344:r
2338:1
2333:r
2330:+
2327:1
2313:2
2310:1
2305:=
2302:z
2285:z
2273:ρ
2261:μ
2246:)
2241:s
2237:)
2228:X
2216:(
2211:n
2203:(
2195:1
2189:n
2185:t
2180:F
2176:=
2173:)
2167:(
2162:t
2158:H
2147:σ
2132:)
2123:)
2114:X
2102:(
2097:n
2089:(
2078:=
2075:)
2069:(
2058:H
2047:σ
2028:2
2023:1
2017:n
1988:2
1983:1
1977:n
1968:F
1957:σ
1940:)
1933:/
1927:2
1923:s
1919:)
1916:1
1910:n
1907:(
1904:(
1897:2
1892:1
1886:n
1877:F
1870:1
1867:=
1864:)
1858:(
1851:2
1842:H
1828:σ
1821:μ
1807:)
1801:(
1796:A
1792:H
1781:μ
1767:)
1761:(
1756:t
1752:H
1741:μ
1737:σ
1723:)
1717:(
1706:H
1695:μ
1687:σ
1649:)
1644:2
1640:s
1636:,
1627:X
1621:(
1618:N
1598:)
1593:2
1585:,
1576:X
1570:(
1567:N
1547:)
1541:(
1536:A
1532:H
1511:)
1505:(
1500:t
1496:H
1485:μ
1481:n
1463:)
1458:s
1454:)
1445:X
1433:(
1428:n
1420:(
1409:=
1406:)
1400:(
1395:A
1391:H
1377:μ
1360:,
1356:)
1351:s
1347:)
1338:X
1326:(
1321:n
1313:(
1305:1
1299:n
1295:t
1290:F
1286:=
1283:)
1277:(
1272:t
1268:H
1257:,
1253:)
1244:)
1235:X
1223:(
1218:n
1210:(
1199:=
1196:)
1190:(
1181:H
1157:)
1151:(
1146:t
1142:H
1121:)
1115:(
1104:H
1081:1
1075:n
1071:t
1046:1
1040:n
1036:t
1031:F
1020:Φ
1012:σ
1004:n
1000:i
996:σ
992:μ
990:(
988:N
983:i
979:X
928:p
922:)
917:p
913:A
903:(
900:P
874:p
870:A
845:C
823:p
819:A
798:C
778:I
758:C
738:)
735:1
732:,
729:0
726:(
720:I
714:p
692:p
688:A
667:1
661:p
655:0
635:p
593:p
589:A
565:p
562:=
559:)
554:p
550:A
546:(
543:C
492:C
461:n
457:H
453:U
445:H
437:.
435:U
430:n
425:X
420:0
417:θ
412:n
407:X
404:(
401:n
397:H
393:0
390:θ
388:(
385:n
381:H
377:0
374:θ
370:θ
365:;
363:Θ
358:n
354:X
352:(
349:n
345:H
340:n
336:H
332:χ
327:n
323:X
316:θ
312:Θ
308:χ
303:n
298:X
295:(
292:n
288:H
283:n
279:H
270:n
266:X
262:1
259:X
254:n
249:X
244:χ
240:θ
236:Θ
220:θ
212:θ
206:.
204:θ
199:n
195:ξ
190:n
186:H
181:n
177:X
173:n
170:X
168:(
165:n
161:ξ
157:α
155:(
152:n
148:ξ
144:θ
140:α
138:(
135:n
131:ξ
127:α
31:(
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.