Estimation of the truncation probability in the random truncation model

Abstract

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 $G_n$.