Empirical processes based on pseudo-observations

Usually, empirical distribution functions are used to estimate the theoretical distribution function of known functions theta(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 theta(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 theta(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.