The asymptotic power function of GLR tests for local change of parameter

Abstract

Retrospective tests are constructed to detect a local shift of parameter of a distribution function occuring at unknown point of time between consecutive independent observations. Three parametric models of distributions are considered; the one parameter exponential family, the location parameter family and the scale parameter family. The class of tests considered is based on the Generalized Likelihood Ratio (GLR) tests, appropriately adapted for such a change-point problem. Asymptotic techniques are used to obtain the limiting distribution of the test statistics, under both the null hypothesis of no change and the change-point alternative. The test statistics, being maximum likelihood type statistics, are shown to converge (in distribution) to the supremum of Brownian motion process, with or without a change-point according to the alternative and the null hypothesis respectively. Analytical expressions for the asymptotic power functions of the proposed tests are provided. These results are then used to provide power comparisons of the tests with those of the Chernoff-Zacks' quasi-Bayesian test.