Expansions for posterior distributions

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
C. Withers, S. Nadarajah
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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.
后验分布的展开式
:设X n是一个大小为n的具有对数似然nl (θ)的样本,其中θ是R p中具有先验分布ξ (θ)的未知参数。我们不需要假设样本值是独立的,甚至是平稳的。设(cid:98) θ为最大似然估计(MLE)。我们证明了θ | X n是渐近正态的,具有均值(cid:98) θ和协方差- n−1 l (cid:5), (cid:5) (cid:16)(cid:98) θ (cid:17)−1,其中l (cid:5), (cid:5) (θ) =∂2 l (θ) /∂θ∂θ '。相反,(cid:98) θ | θ渐近正态,均值θ和协方差n−1 [I (θ)]−1,其中I (θ) =−E (cid:104) l (cid:5), (cid:5) (cid:16)(cid:98) θ (cid:17) | θ (cid:105)为Fisher信息。所以,以θ为条件的频率推理不能用来近似贝叶斯推理,除了指数族。然而,在温和条件下,−1 (cid:5), (cid:5) (cid:16)(cid:98) θ (cid:17) | θ→I (θ)的概率。因此,贝叶斯推理(即以X n为条件)可以用来近似频率推理。对于t (θ)任意光滑函数,我们得到了L t (θ) |xn的后向累积展开式、后向Edgeworth- cornish - fisher (ECF)展开式和后向倾斜Edgeworth展开式,以及t (θ) |xn高精度的置信区域。我们还给出了t (θ)关于t (cid:16)(cid:98) θ (cid:17)的贝叶斯估计(估计量)的展开式,以及关于(cid:98) θ的最大后验估计,以及它们相对于平方误差损失的相对效率。
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来源期刊
CiteScore
1.60
自引率
10.00%
发文量
30
审稿时长
>12 weeks
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. More specifically, the following types of contributions will be considered: (i) Original articles dealing with methodological developments, comparison of competing techniques or their computational aspects. (ii) Original articles developing theoretical results. (iii) Articles that contain novel applications of existing methodologies to practical problems. For these papers the focus is in the importance and originality of the applied problem, as well as, applications of the best available methodologies to solve it. (iv) Survey articles containing a thorough coverage of topics of broad interest to probability and statistics. The journal will occasionally publish book reviews, invited papers and essays on the teaching of statistics.
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