The central limit theorem under random truncation

Details The central limit theorem under random truncation The central limit theorem under random truncation

2008-10-22

arxiv.org/abs/0810.3985v1

Bernoulli 2008, Vol. 14, No. 3, 604-622

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/07-BEJ116 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/07-BEJ116

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 $\int\varphi \mathrm{d}F_n$ of the nonparametric maximum likelihood estimator (MLE) of $F$, the Lynden-Bell estimator $F_n$. A useful representation of $\int \varphi \mathrm{d}F_n$ is derived which yields asymptotic normality under optimal moment conditions on the score function $\varphi$. 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.

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