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June 1998 The strong law under random truncation
Shuyuan He, Grace L. Yang
Ann. Statist. 26(3): 992-1010 (June 1998). DOI: 10.1214/aos/1024691085


The random truncation model is defined by the conditional probability distribution $H (x, y) =P[X\leq x,Y\leq y |X \geq Y] where $X$ and $Y$ are independent random variables. A problem of interest is the estimation of the distribution function $F$ of $X$ with data from the distribution $H$. Under random truncation, $F$ need not be fully identifiable from $H$ and only a part of it, say $F_0$ , is. We show that the nonparametric MLE $F_n$ of $F_0$ obeys the strong law of large numbers in the sense that for any nonnegative, measurable function $\phi(x)$, the integrals $\int\phi(x)dF_n(x)\to\int\phi(x)dF_0(x)$ almost surely as $n$ tends to infinity. Similar results were first obtained by Stute and Wang for the right censoring model. The results are useful in establishing the strong consistency of various estimates. Some of our results are derived from the weak consistency of $F_n$ obtained by Woodroofe.


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Shuyuan He. Grace L. Yang. "The strong law under random truncation." Ann. Statist. 26 (3) 992 - 1010, June 1998.


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

zbMATH: 0929.62037
MathSciNet: MR1635430
Digital Object Identifier: 10.1214/aos/1024691085

Primary: 62G20

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.26 • No. 3 • June 1998
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