Uniform Representation of Product‐Limit Integrals with Applications

Abstract

Abstract. 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.