The strong law under random truncation

Abstract

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.