A note on martingale transforms for model checks

Abstract

Martingale transforms are a well known tool to derive asymptotically distribution free tests for statistics based on empirical processes. Since its introduction by Khmaladze (1981) they have been frequently applied to many testing problems. In this paper martingale transforms for empirical processes are discussed from a non standard perspective with a specific focus on the case where the null hypothesis is not satisfied. For the sake of a transparent presentation we restrict our investigations to the problem of checking model assumptions in regression models, but the conclusions are generally valid. We show the weak convergence of empirical processes under fixed alternatives and introduce a new version of the martingale transform such that the transformed limiting process is a Brownian motion in scaled time, even if the null hypothesis is not satisfied.