Uniform Representation of Product-Limit Integrals with Applications

Let "X" be a "d"-variate random vector that is completely observed, and let "Y" be a random variable that is subject to right censoring and left truncation. For arbitrary functions "ϕ" we consider expectations of the form "E"["ϕ"("X", "Y")], which appear in many statistical problems, and we estimate these expectations by using a product-limit estimator for censored and truncated data, extended to the context where covariates are present. An almost sure representation for these estimators is obtained, with a remainder term that is of a certain negligible order, uniformly over a class of "ϕ"-functions. This uniformity is important for the application to goodness-of-fit testing in regression and to inference for the regression depth, which we consider in more detail. Copyright 2005 Board of the Foundation of the Scandinavian Journal of Statistics..