Uniform Representation of Product‐Limit Integrals with Applications

César Sánchez Sellero, Wenceslao González Manteiga and Ingrid Van Keilegom ()

Scandinavian Journal of Statistics, 2005, vol. 32, issue 4, 563-581

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.

Date: 2005
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