Empirical Processes Based on Pseudo - Observations 3

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 Q n , the quantities (Q n ; 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 pseudo-observations will be given together with applications to copulas, weighted symmetry, regression and other statistical concepts. 1. Introduction In a regression setting, including the study of time series, the series of residuals is used to construct empirical distribution functions for goodness-of-t tests, for prediction intervals, and so on. Empirical processes built from residuals received considerable attention lately; In a more general setting, consider the problem of approximating 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 Q n , the quantities (Q n ; X i) are called pseudo-observations; obviously they are a generalization of residuals. In Genest and Rivest 1993], pseudo-observations were used to approximate the distribution function of H(X), where H is the distribution function of the bivariate random vector X = (X (1) ; X (2)). Their estimation procedure was the following: take a random sample