Semiparametric estimation of the occurrence rate for prevalent survival data in competing risk models

Abstract

In many follow-up studies, the problem of interest is the estimation of occurrence probabilities based on prevalent data in competing risks models. Let and denote the lifetimes of two competing risks that can be dependent on each other. Let V* and C* denote left truncation and right censoring variables, respectively. Assume that (, ) and (V*, C*) are independent of each other but V* and C* are dependent with P(C*≥V*) = 1. For left-truncated and right-censored data, one can observe nothing if , and observe (X*, δ*), if Z*≥V*, where X* = min (Z*, C*) and δ* is equal to one if , equal to two if and zero otherwise. Let π₁ (x) and π₂(x) denote the occurence probability of and , respectively. Huang and Wang [Y. Huang and M.-C. Wang, Estimating the occurrence of rate for prevalent survival date in competing risk models, J. AM. Statist. Assoc. 90(432) (1995), pp. 1406–1415] derive the maximum likelihood estimates of π _k (x) (k = 1, 2) under nonparametric and length-biased (i.e. V* is uniformly distributed and C* = ∞) models. In this article, we extend previous models by considering the case when the distribution of V* is parameterized as G(x; θ) and the distributions of and C* are left unspecified. Several iterative algorithms are proposed to obtain the semiparametric estimates (denoted by π _k (x; θˆ _n ) of π _k (x) (k = 1, 2). The asymptotic properties of π _k (x; θˆ _n ) are discussed. The simulation results show that the semiparametric estimators π _k (x; θˆ _n ) have smaller mean-squared errors compared to the nonparametric estimators of π _k (k = 1, 2).

Keywords:

• EM algorithm
• maximum likelihood estimation
• occurrence probabilities