Semiparametric Likelihood Ratio-Based Inferences for Truncated Data

Abstract

In astronomic, demographic, epidemiologic, and other studies, the variable of interest, say the survival time, is often truncated by an associated variable. In many situations, the distribution of the truncation variable can be described by a parametric form. Unlike in the standard right-censorship model in which the censoring distribution is noninformative, knowledge of the truncation distribution can be used to improve estimation of the survival distribution. This article derives likelihood ratio-based confidence intervals for survival probabilities and for the truncation proportion under the two models in which the truncation distribution is assumed either to be known or to belong to a parametric family. Our proposed methods enable one to incorporate both the information contained in the data and the available information on the truncation distribution and thus are expected to have better performance than fully nonparametric methods. Our approach also has applications to some biased sampling problems. A simulation study is done to assess the small-sample performance of the proposed methods and to compare it with some existing nonparametric methods. An illustration is also given using a transfusion-related acquired immune deficiency syndrome (AIDS) data.

Key Words:

• Biased sampling
• Empirical likelihood
• Incubation time
• Interval estimation
• Nonparametric maximum likelihood estimate