Asymptotic Properties of the Product Limit Estimate Under Random Truncation
Mei-Cheng Wang, Nicholas P. Jewell, Wei-Yann Tsai
The Annals of Statistics · 1986 · Volume 14 · Issue 4
Abstract
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.
Keywords
62G05, covariance structure, Product limit, truncated data, weak convergence
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