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

In this paper we study stochastic processes which enable monitoring the possible changes of probability distributions over time. These so-called monitoring processes are bivariate functions of time and position at the measurement scale, and in particular be used to test the null hypothesis of no change: one may then form Kolmogorov--Smirnov or other type of tests as functionals of the processes. In Hjort and Koning (2001) Cram??r-type deviation results were obtained under the constancy null hypothesis for [bootstrapped versions of] such ``derived'' test statistics. Here the behaviour of derived test statistics is investigated under alternatives in the vicinity of the constancy hypothesis. When combined with Cram??r-type deviation results, the results in this paper enable the computation of efficiencies of the corresponding tests. The discussion of some examples of yield guidelines for the choice of the test statistic, and hence for the underlying monitoring process.