Empirical Processes Based on Pseudo-Observations

Abstract

. Usually, empirical distribution functions are used to estimate the theoretical distribution function of known functions `(X) of the observable random variable X. In practice, many researchers are using empirical distribution functions constructed from residuals, which are estimations of a non-observable error terms in linear models. This falls under a class of more general problems in which one is interested in the estimation of the distribution function of a non-observable random variable `(Q; X) depending on an observable random variable X together with its unknown law Q. When Q is estimated by some Qn , the quantities `(Qn ; X i ) are called pseudo-observations. Some work has been done recently when the pseudo-observations are the so-called residuals of linear models. The aim of this paper is to provide some tools to study the asymptotic behavior of empirical processes constructed from general pseudo-observations. Examples of pseudoobservations will be given together with applica..