The Annals of Statistics

Estimation of the truncation probability in the random truncation model

Shuyuan He and Grace L. Yang

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Under random truncation, a pair of independent random variables $X$ and $Y$ is observable only if $X$ is larger than $Y$. The resulting model is the conditional probability distribution $H( x, y) =P[X \leq x,Y \leq y|X \geq Y]$. For the truncation probability $\alpha=P[X \geq Y]$, a proper estimate is not the sample proportion but $\alpha_n=\int G_n (s)dF_n(s)$ where $F_n$ and $G_n$ are product limit estimates of the distribution functions $F$ and$G$ of $X$ and$Y$, respectively. We obtain a much simpler representation $\hat {\alpha}_n$ for $\alpha_n$. With this, the strong consistency, an iid representation (and hence asymptotic normality), and a LIL for the estimate are established. The results are true for arbitrary$F$ and $G$. The continuity restriction on $F$ and $G$ often imposed in the literature is not necessary. Furthermore, the representation $\hat {\alpha}_n$ of $\alpha_n$ facilitates the establishment of the strong law for the product limit estimates $F_n$ and $G_n$.

Article information

Ann. Statist., Volume 26, Number 3 (1998), 1011-1027.

First available in Project Euclid: 21 June 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05.

Random truncation nonparametric estimation product-limit truncation probability strong consistency LIL iid representation


He, Shuyuan; Yang, Grace L. Estimation of the truncation probability in the random truncation model. Ann. Statist. 26 (1998), no. 3, 1011--1027. doi:10.1214/aos/1024691086.

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