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

摘要: 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 natural parameter $\phi_\partial(\bar{X}_n)$ $\log$ likelihood ratio statistic for testing null hypothesis $\{\partial\}$. The technique involves approximating convex compact sets in $R^k$ by polytopes, then estimating contents associated closed halfspaces, counting number these half-spaces. Some examples are given, among them multivariate normal distribution with unknown mean vector covariance matrix.