A Large Deviation Result for the Likelihood Ratio Statistic in Exponential Families

Abstract

In this paper we consider exponential families of distributions and obtain under certain conditions a uniform large deviation result about the tail probability $P_\partial(\phi_\partial(\bar{X}_n) > \varepsilon), \varepsilon > 0$, where $\partial$ is the natural parameter and $\phi_\partial(\bar{X}_n)$ is the $\log$ likelihood ratio statistic for testing the null hypothesis $\{\partial\}$. The technique involves approximating certain convex compact sets in $R^k$ by polytopes, then estimating the probability contents of associated closed halfspaces, and counting the number of these half-spaces. Some examples are given, among them the multivariate normal distribution with unknown mean vector and covariance matrix.