The central limit theorem under random truncation

Under left truncation, data (X(i), Y(i)) are observed only when Y(i) <= X(i). Usually, the distribution function F of the X(i) is the target of interest. In this paper, we study linear functionals integral phi dF(n) of the nonparametric maximum likelihood estimator (MLE) of F, the Lynden-Bell estimator F(n-) A useful representation of integral phi dF(n) is derived which yields asymptotic normality under optimal moment conditions on the score function W. 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.