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 phi we consider expectations of the form E[phi(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 phi-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.