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October 2002 Asymptotic normality with small relative errors of posterior probabilities of half-spaces
R. M. Dudley, D. Haughton
Ann. Statist. 30(5): 1311-1344 (October 2002). DOI: 10.1214/aos/1035844978

Abstract

Let $\Theta$ be a parameter space included in a finite-dimensional Euclidean space and let $A$ be a half-space. Suppose that the maximum likelihood estimate $\theta_n$ of $\theta$ is not in $A$ (otherwise, replace $A$ by its complement) and let $\Delta$ be the maximum log likelihood (at $\theta_n$) minus the maximum log likelihood over the boundary $\partial A$. It is shown that under some conditions, uniformly over all half-spaces $A$, either the posterior probability of $A$ is asymptotic to $\Phi(-\sqrt{2\Delta}\,)$ where $\Phi$ is the standard normal distribution function, or both the posterior probability and its approximant go to 0 exponentially in $n$. Sharper approximations depending on the prior are also defined.

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R. M. Dudley. D. Haughton. "Asymptotic normality with small relative errors of posterior probabilities of half-spaces." Ann. Statist. 30 (5) 1311 - 1344, October 2002. https://doi.org/10.1214/aos/1035844978

Information

Published: October 2002
First available in Project Euclid: 28 October 2002

zbMATH: 1014.62031
MathSciNet: MR1936321
Digital Object Identifier: 10.1214/aos/1035844978

Subjects:
Primary: 62F15
Secondary: 60F99 , 62F05

Keywords: Bernstein-von Mises theorem , gamma tail probabilities , intermediate deviations , Jeffreys prior , Mills' ratio

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 5 • October 2002
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