Computationally simple estimation and improved efficiency for special cases of double truncation

Doubly truncated survival data arise when event times are observed only if they occur within subject specific intervals of times. Existing iterative estimation procedures for doubly truncated data are computationally intensive (Turnbull 38:290-295, 1976; Efron and Petrosian 94:824-825, 1999; Shen 62:835-853, 2010a). These procedures assume that the event time is independent of the truncation times, in the sample space that conforms to their requisite ordering. This type of independence is referred to as quasi-independence. In this paper we identify and consider two special cases of quasi-independence: complete quasi-independence and complete truncation dependence. For the case of complete quasi-independence, we derive the nonparametric maximum likelihood estimator in closed-form. For the case of complete truncation dependence, we derive a closed-form nonparametric estimator that requires some external information, and a semi-parametric maximum likelihood estimator that achieves improved efficiency relative to the standard nonparametric maximum likelihood estimator, in the absence of external information. We demonstrate the consistency and potentially improved efficiency of the estimators in simulation studies, and illustrate their use in application to studies of AIDS incubation and Parkinson's disease age of onset.