{"title":"Expansions for posterior distributions","authors":"C. Withers, S. Nadarajah","doi":"10.1214/22-bjps561","DOIUrl":null,"url":null,"abstract":": Suppose that X n is a sample of size n with log likelihood nl ( θ ), where θ is an unknown parameter in R p having a prior distribution ξ ( θ ). We need not assume that the sample values are independent or even stationary. Let (cid:98) θ be the maximum likelihood estimate (MLE). We show that θ | X n is asymptotically normal with mean (cid:98) θ and covariance − n − 1 l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) − 1 , where l (cid:5) , (cid:5) ( θ ) = ∂ 2 l ( θ ) /∂θ∂θ ′ . In contrast (cid:98) θ | θ is asymptotically normal with mean θ and covariance n − 1 [ I ( θ )] − 1 , where I ( θ ) = − E (cid:104) l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) | θ (cid:105) is Fisher’s information. So, frequentist inference conditional on θ cannot be used to approximate Bayesian inference, except for exponential families. However, under mild conditions − l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) | θ → I ( θ ) in probability. So, Bayesian inference (that is, conditional on X n ) can be used to approximate frequentist inference. For t ( θ ) any smooth function, we obtain posterior cumulant expansions, posterior Edgeworth-Cornish-Fisher (ECF) expansions and posterior tilted Edgeworth expansions for L t ( θ ) | X n , as well as confidence regions for t ( θ ) | X n of high accuracy. We also give expansions for the Bayes estimate (estimator) of t ( θ ) about t (cid:16)(cid:98) θ (cid:17) , and for the maximum a posteriori estimate about (cid:98) θ , as well as their relative efficiencies with respect to squared error loss.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Brazilian Journal of Probability and Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/22-bjps561","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
: Suppose that X n is a sample of size n with log likelihood nl ( θ ), where θ is an unknown parameter in R p having a prior distribution ξ ( θ ). We need not assume that the sample values are independent or even stationary. Let (cid:98) θ be the maximum likelihood estimate (MLE). We show that θ | X n is asymptotically normal with mean (cid:98) θ and covariance − n − 1 l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) − 1 , where l (cid:5) , (cid:5) ( θ ) = ∂ 2 l ( θ ) /∂θ∂θ ′ . In contrast (cid:98) θ | θ is asymptotically normal with mean θ and covariance n − 1 [ I ( θ )] − 1 , where I ( θ ) = − E (cid:104) l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) | θ (cid:105) is Fisher’s information. So, frequentist inference conditional on θ cannot be used to approximate Bayesian inference, except for exponential families. However, under mild conditions − l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) | θ → I ( θ ) in probability. So, Bayesian inference (that is, conditional on X n ) can be used to approximate frequentist inference. For t ( θ ) any smooth function, we obtain posterior cumulant expansions, posterior Edgeworth-Cornish-Fisher (ECF) expansions and posterior tilted Edgeworth expansions for L t ( θ ) | X n , as well as confidence regions for t ( θ ) | X n of high accuracy. We also give expansions for the Bayes estimate (estimator) of t ( θ ) about t (cid:16)(cid:98) θ (cid:17) , and for the maximum a posteriori estimate about (cid:98) θ , as well as their relative efficiencies with respect to squared error loss.
期刊介绍:
The Brazilian Journal of Probability and Statistics aims to publish high quality research papers in applied probability, applied statistics, computational statistics, mathematical statistics, probability theory and stochastic processes.
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