Under random truncation, a pair of independent random variables U* and V* is observable only if U* is larger than V*. The resulting model is the conditional probability .For the truncation probability ,a proper estimate is , where and are nonparametric maximum likelihood estimate (NPMLE) of the distributions F and G. He and Yang (1998) showed that is equivalent to a simpler representation . In this article, using coupled inverse-probability-of-truncation weighted estimators, we propose an alternative proof of the equivalence. Similarly, for left-truncated and right-censored data, two estimators (denoted by and ) are considered. It is shown that the equivalence of and does not hold.Simulation results shows that the mean-squared error of is smaller than that of . Under random truncation, a pair of independent random variables U* and V* is observable only if U* is larger than V*. The resulting model is the conditional probability .For the truncation probability ,a proper estimate is , where and are nonparametric maximum likelihood estimate (NPMLE) of the distributions F and G. He and Yang (1998) showed that is equivalent to a simpler representation . In this article, using coupled inverse-probability-of-truncation weighted estimators, we propose an alternative proof of the equivalence. Similarly, for left-truncated and right-censored data, two estimators (denoted by and ) are considered. It is shown that the equivalence of and does not hold.Simulation results shows that the mean-squared error of is smaller than that of .
