Estimation from cross-sectional data under a semiparametric truncation model

Cross-sectional sampling is often used when investigating inter-event times, resulting in left-truncated and right-censored data. In this paper, we consider a semiparametric truncation model in which the truncating variable is assumed to belong to a certain parametric family. We examine two methods of estimating both the truncation and the lifetime distributions. We obtain asymptotic representations of the estimators for the lifetime distribution and establish their weak convergence. Both of the proposed estimators perform better than Wang’s (1991) nonparametric maximum likelihood estimator in terms of the integrated mean squared error, when the parametric family for the truncation is sufficiently close to its true distribution. The full likelihood approach is preferable to the conditional likelihood approach in estimating the lifetime distribution, though not necessarily the truncation distribution. In an application to Alzheimer’s disease data, hypothesis tests reject the uniform truncation distribution, but several other parametric models lead to similar behaviour of the truncation and lifetime distributions after disease onset.