A Berry-Esséen Bound for the Product-Limit Estimator under Left-Truncation and Right-Censoring

Abstract

For left-truncated and right-censored data, the product-limit estimator Fˆ_xis a well-known nonparametric estimator for the distribution function F_x of the target variable X such as the survival time. Since Fˆ_xas a very complicated product form we establish first the Berry-Esseen bound for the cumulative hazard estimator of F_x The cumulative hazard estimator can be represented as a U-statistic. By using the result in Helmers and van Zwet [6], we derive the Berry-esséen bound for this U-statistic. Then Berry-Esseen bounds for the distribution of the cumulative hazard estimator and the normal distribution and the distribution of the product-limit estimator and the normal distribution are obtained.

Keywords:

• Berry-Esséen bound
• truncation
• censoring
• U-statistic
• product-limit estimator

AMS:

• 62N01
• 62N02