On the rate of approximations for maximum likelihood tests in change-point models

We study the asymptotics of maximum-likelihood ratio-type statistics for testing a sequence of observations for no change in parameters against a possible change while some nuisance parameters remain constant over time. We obtain extreme value as well as Gaussian-type approximations for the likelihood ratio. We get necessary and sufficient conditions for the weak convergence of supremum and L(p)-functionals of the likelihood ration process. We also approximate the maximum Likelihood ratio with Omstein-Uhlenbeck processes and obtain bounds for the rate of approximation. We show that the Ornstein-Uhlenbeck approach is superior to the extreme value limit in case of moderate sample sizes. (C) 1996 Academic Press, Inc.