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The central limit theorem under random truncation.

Winfried Stute¹, Jane-Ling Wang

¹Mathematical Institute, University of Giessen, Arndtstr. 2, D-35392 Giessen, Germany. winfried.stute@math.uni-giessen.de.

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|July 31, 2012

Summary

This summary is machine-generated.

This study analyzes the Lynden-Bell estimator for left-truncated data, deriving asymptotic normality for linear functionals. The findings apply without assuming data continuity, improving statistical inference for truncated distributions.

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Area of Science:

Statistics
Survival Analysis
Nonparametric Inference

Background:

Left-truncated data, where observations are only available when Y ≤ X, is common in survival analysis.
Estimating the underlying distribution function F from such data is crucial for accurate analysis.
The Lynden-Bell estimator is a key tool for nonparametric estimation under truncation.

Purpose of the Study:

To derive a useful representation for linear functionals of the Lynden-Bell estimator (F(n)).
To establish asymptotic normality for these functionals under optimal moment conditions.
To investigate the distributional convergence of the Lynden-Bell empirical process.

Main Methods:

Derivation of a novel representation for linear functionals of the Lynden-Bell estimator.
Application of asymptotic theory to establish normality.
Analysis of the Lynden-Bell empirical process without continuity assumptions.

Main Results:

A useful representation of linear functionals ∫ φ dF(n) was derived.
Asymptotic normality was established under optimal moment conditions for the score function φ.
Distributional convergence of the Lynden-Bell empirical process was obtained on the entire real line.

Conclusions:

The derived representation and asymptotic normality provide powerful tools for statistical inference with left-truncated data.
The results hold without requiring the continuity of the underlying distribution F.
This work advances the understanding and application of the Lynden-Bell estimator in survival analysis.