The Annals of Statistics

Asymptotic Properties of the Product Limit Estimate Under Random Truncation

Mei-Cheng Wang, Nicholas P. Jewell, and Wei-Yann Tsai

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Many authors have considered the problem of estimating a distribution function when the observed data is subject to random truncation. A prominent role is played by the product limit estimator, which is the analogue of the Kaplan-Meier estimator of a distribution function under random censoring. Wang and Jewell (1985) and Woodroofe (1985) independently proved consistency results for this product limit estimator and showed weak convergence to a Gaussian process. Both papers left open the exact form of the covariance structure of the limiting process. Here we provide a precise description of the asymptotic behavior of the product limit estimator, including a simple explicit form of the asymptotic covariance structure, which also turns out to be the analogue of the covariance structure of the Kaplan-Meier estimator. Some applications are briefly discussed.

Article information

Ann. Statist., Volume 14, Number 4 (1986), 1597-1605.

First available in Project Euclid: 12 April 2007

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Primary: 62G05: Estimation

Product limit truncated data weak convergence covariance structure


Wang, Mei-Cheng; Jewell, Nicholas P.; Tsai, Wei-Yann. Asymptotic Properties of the Product Limit Estimate Under Random Truncation. Ann. Statist. 14 (1986), no. 4, 1597--1605. doi:10.1214/aos/1176350180.

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