A generalized product-limit estimator for truncated data

Abstract

A nonparametric maximum likelihood estimator (NPMLE) of the underlying distribution in the presence of truncation, first published in the astronomy literature by Lynden-Bell in 1971, has received considerable attention in the statistics and biostatistics literature. A limitation of this useful estimator is that it relies on the independence of the variable of interest and of the truncation variable. In this paper, a generalization of this estimator is proposed, which expands the applicability of the estimator by allowing for dependence between the variable of interest and the truncation variable in the presence of a covariate. Weak convergence and asymptotic optimality in the Hajek–Beran sense (Beran, R, 1977, Estimating a distribution function. Annals of Statistics, 5, 400–404.) are shown for this generalized estimator process. A related estimator of the joint distribution of the covariate and the variable of interest is introduced, which shares the asymptotic normality and optimality of the previous estimator. Finally, simulation results are used to compare the generalization to the Lynden-Bell estimator in a variety of situations.