Uniform representation of product-limit integrals with applications

Sellero, CS;Manteiga, WG;Van Keilegom, Ingrid
(2005) Scandinavian Journal of Statistics : theory and applications — Vol. 32, n° 4, p. 563-581 (2005)

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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 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.
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Sellero, C., Manteiga, W., & Van Keilegom, I. (2005). Uniform representation of product-limit integrals with applications. Scandinavian Journal of Statistics : theory and applications, 32(4), 563-581. https://doi.org/10.1111/j.1467-9469.2005.00453.x (Original work published 2005)