## The Annals of Statistics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 62F03: Hypothesis testing 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 52A25

**Keywords**

Large deviations exponential families convexity polytope

#### Citation

Kourouklis, Stavros. A Large Deviation Result for the Likelihood Ratio Statistic in Exponential Families. Ann. Statist. 12 (1984), no. 4, 1510--1521. doi:10.1214/aos/1176346807. https://projecteuclid.org/euclid.aos/1176346807