The central limit theorem under random truncation

TL;DR

This paper establishes the asymptotic normality of linear functionals of the Lynden-Bell estimator for the distribution function under left truncation, without requiring continuity of the true distribution.

Contribution

It provides a new representation for linear functionals of the Lynden-Bell estimator, enabling asymptotic normality results under minimal moment conditions.

Findings

Asymptotic normality of linear functionals of the Lynden-Bell estimator.

Distributional convergence of the Lynden-Bell empirical process on the entire real line.

No continuity assumption needed on the true distribution.

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 $\int\varphi \mathrm{d}F_n$ 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.