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September, 1977 Approximate Behavior of the Posterior Distribution for a Large Observation
Glen Meeden, Dean Isaacson
Ann. Statist. 5(5): 899-908 (September, 1977). DOI: 10.1214/aos/1176343946

Abstract

Let $X$ be a real valued random variable with a family of possible distributions belonging to a one parameter exponential family with the natural parameter $\theta \in(\underline{\theta}, + \infty)$. Let $g$ be a prior probability density for $\theta$ with unbounded support. Under some additional assumptions it is shown that for large values of $x$ the posterior distribution of $\theta$ given $X = x$ is approximately normally distributed about its mode. If $\delta_g$ denotes the Bayes estimator for squared error loss of some function $\gamma(\theta)$ against $g$ then the rate at which $\delta_g(x)$ approaches infinity as $x$ approaches infinity is found. The rate is shown to depend on the behavior of the prior density $g(\theta)$ for large values of $\theta$.

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Glen Meeden. Dean Isaacson. "Approximate Behavior of the Posterior Distribution for a Large Observation." Ann. Statist. 5 (5) 899 - 908, September, 1977. https://doi.org/10.1214/aos/1176343946

Information

Published: September, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0368.62003
MathSciNet: MR448683
Digital Object Identifier: 10.1214/aos/1176343946

Subjects:
Primary: 62C10
Secondary: 62F10

Keywords: Bayes estimation , exponential family , normal distribution , posterior distribution , quadratic loss

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 5 • September, 1977
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