Weighted Approximations of Parameters-Estimated Empirical Processes and Changepoint Analysis

Abstract

This chapter studies the weighted asymptotic behavior of empirical processes based on independent observations from parametric families and defined after the unknown parameters are estimated by suitably chosen estimators. This chapter derives the limiting Gaussian processes under the null hypothesis of the observations being identically distributed. An application to testing for a change in the distribution at an unknown point of a random sequence is considered. This brings to the notion of defining a two-time parameter sequential version of the estimated empirical process. The results can provide applications to testing for a change in the distribution of a random sequence at an unknown point that combines the nonparametric results for empirical processes with the additional parametric information coming from the estimation. Considering these normal empirical processes, which are of independent interest, has helped to establish the kind of conditions needed in the case where the underlying distribution belongs to a more general parametric family. Indeed, one obtained results along these lines under certain regularity conditions for the family, which enabled one to have conditions for the type of estimation to be used. These results are outlined in this paper. The applications to testing for a change in the distribution at an unknown point can be thus generalized to these multivariate normal observations as well.