The strong law under random truncation

The random truncation model is defined by the conditional probability distr ibution H(x, y) = P[X less than or equal to x, Y less than or equal to y\X greater than or equal to 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 integra ls integral phi(x) dF(n)(x) --> integral phi(x) dF(0)(x) almost surely as n tends to infinity. Similar results were first obtained by Stute and Wang f or the right censoring model. The results are useful in establishing the st rong consistency of various estimates. Some of our results are derived from the weak consistency of F-n obtained by Woodroofe.