The central limit theorem under random truncation

Abstract

Under left truncation, data (Xi, Yi) are observed only when Yi ≤ Xi. Usually, the distribution function F of the Xi is the target of interest. In this paper, we study linear functionals ∫ dFn of the nonparametric maximum likelihood estimator (MLE) of F, the Lynden-Bell estimator Fn. A useful representation of ∫ dFn is derived which yields asymptotic normality under optimal moment conditions on the score function. No continuity assumption on F is required. As a by-product, we obtain the distributional convergence of the Lynden-Bell empirical process on the whole real line. © 2008 ISI/BS.