The central limit theorem under random truncation

Abstract

Under left truncation, data $(X_i,Y_i)$ are observed only when $Y_i\le X_i$. Usually, the distribution function $F$ of the $X_i$ is the target of interest. In this paper, we study linear functionals $\int \phi \mathrm{d}{F}_n$ of the nonparametric maximum likelihood estimator (MLE) of $F$, the Lynden-Bell estimator $F_n$. A useful representation of $\int \phi \mathrm{d}{F}_n$ is derived which yields asymptotic normality under optimal moment conditions on the score function $\phi$. 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.

Published in at http://dx.doi.org/10.3150/07-BEJ116 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)