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June 1998 Estimation of the truncation probability in the random truncation model
Shuyuan He, Grace L. Yang
Ann. Statist. 26(3): 1011-1027 (June 1998). DOI: 10.1214/aos/1024691086


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 $G_n$.


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Shuyuan He. Grace L. Yang. "Estimation of the truncation probability in the random truncation model." Ann. Statist. 26 (3) 1011 - 1027, June 1998.


Published: June 1998
First available in Project Euclid: 21 June 2002

zbMATH: 0929.62036
MathSciNet: MR1635434
Digital Object Identifier: 10.1214/aos/1024691086

Primary: 62G05.

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.26 • No. 3 • June 1998
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