| Abstract: | Let T*, C* and V* denote the lifetime, censoring and truncation variables, respectively. Assume that (C*, V*) is independent of T* and P(C*>=V*) = 1. Let F, Q and G denote the common distribution function of T*, C* and V*, respectively. For left-truncated and right-censored (LTRC) data, one can observe nothing if T*<V* and observe (X*, .delta.*), with X*= min(T*, C*) and .delta.* = I╱sub╱[T*<=C*], if T*>=V*. For LTRC data, the truncation productlimit estimate Fn is the maximum likelihood estimate (MLE) for nonparametric models. If the distribution of V* is parameterized as G(x; .theta.) and the distributions of T* and C* are left unspecified, the product-limit estimate Fn is not the MLE for this semiparametric model. For left-truncated data, Wang (1989) derived the MLE of F for the semiparametric model and established its weak convergence properties. In this note, for LTRC data, two semiparametric estimates, Fn(x; .theta.n) and Fu(x; .theta.), are proposed for the semiparametric model. A simulation study is conducted to compare the performances of the two semiparametric estimators against that of Fn(x). |
