## The Annals of Statistics

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

Stavros Kourouklis

#### 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.

#### Article information

Source
Ann. Statist., Volume 12, Number 4 (1984), 1510-1521.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346807

Digital Object Identifier
doi:10.1214/aos/1176346807

Mathematical Reviews number (MathSciNet)
MR760703

Zentralblatt MATH identifier
0551.62017

JSTOR