A Semiparametric Model for Randomly Truncated Data

Abstract For randomly censored data, it is known that the maximum likelihood estimate (MLE) of the survival curve is not affected by parametric assumption on the censoring variable. The Kaplan-Meier (1958) estimate is the MLE for both nonparametric and semiparametric models. For randomly truncated data, the truncation product-limit estimate is the MLE for nonparametric models. This is not the case if the truncation mechanism is parameterized, however. Specifically, let X be a generic random variable and T be the truncation variable. If the distribution of T is parameterized and the distribution of X is left unspecified, it can be shown that the truncation product-limit estimate is not the MLE for this semiparametric model, even though it is for the fully nonparametric model. In this article the MLE is characterized for the semiparametric model, and the large-sample properties of the estimate are established. The results show that, unlike censoring, the parametric information from the truncation mechanism influences the estimation procedures. Several examples in biostatistics, in which the truncation distribution can be interpreted as the infection distribution or the birth distribution, are considered.