Constancy of distributions: asymptotic efficiency of certain nonparametric tests of constancy

Abstract

In this paper we study stochastic processes which enable monitoring thepossible changes of probability distributions over time. Theseso-called monitoring processes are bivariate functions of time andposition at the measurement scale, and in particular be used to testthe null hypothesis of no change: one may then form Kolmogorov--Smirnovor other type of tests as functionals of the processes. In Hjort andKoning (2001) Cram??r-type deviation results were obtained under theconstancy null hypothesis for [bootstrapped versions of] such``derived'' test statistics. Here the behaviour of derived test statistics is investigated underalternatives in the vicinity of the constancy hypothesis. Whencombined with Cram??r-type deviation results, the results in thispaper enable the computation of efficiencies of the correspondingtests. The discussion of some examples of yield guidelines for thechoice of the test statistic, and hence for the underlying monitoringprocess.Asymptotic efficiency;Constancy of distributions;Empirical distribution functions;Kernel density estimator