Detection of random change point in one-parameter exponential families

Abstract

We consider a sequence X 1, X 2, …, X n of independent random variables which are susceptible to changing their distribution after the [ nT] first observations where T is a random variable of a distribution known with support in ]0, 1[. The object of this work is to detect the eventual change of distribution—for that—we study the performance of a test based on the statistic of Log-likelihood. It shows that when the number of observations gets larger, the distribution of the statistic Log-likelihood behaves like that of an affine function of the Brownian motion; this allows, by using the concept of contiguity in the sense of LeCam, the evaluation of the asymptotic power function of the test.