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 $\theta$.
Publié le : 1977-09-14
Classification:
Bayes estimation,
exponential family,
quadratic loss,
posterior distribution,
normal distribution,
62C10,
62F10
@article{1176343946,
author = {Meeden, Glen and Isaacson, Dean},
title = {Approximate Behavior of the Posterior Distribution for a Large Observation},
journal = {Ann. Statist.},
volume = {5},
number = {1},
year = {1977},
pages = { 899-908},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343946}
}
Meeden, Glen; Isaacson, Dean. Approximate Behavior of the Posterior Distribution for a Large Observation. Ann. Statist., Tome 5 (1977) no. 1, pp. 899-908. http://gdmltest.u-ga.fr/item/1176343946/