Approximations of the Empirical Process when Parameters are Estimated

Abstract

Almost sure and in-probability representations of the empirical process by appropriate Gaussian processes are obtained when unknown parameters of the underlying distribution function are estimated. As to the method of estimation, we consider maximum likelihood and maximum likelihood-like estimators and construct the above-mentioned representations under a null hypothesis. Similar results are obtained also when using Durbin's more general class of estimators under a sequence of alternatives which converge to the null hypothesis. The resulting Gaussian processes depend, in general, on the true value of the unknown parameters.