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Asymptotic properties of the product limit estimate under random truncation. (English) Zbl 0656.62048
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. The first two authors [Tech. Rep., Program in Biostatistics, Univ. California, Berkeley (1985)] and M. Woodroofe [ibid. 13, 163-177 (1985; Zbl 0574.62040)] 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.

MSC:
62G05 Nonparametric estimation
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