A general semiparametric model for left-truncated and right-censored data

Abstract

In many follow-up studies, survival data are often observed according to a cross-sectional sampling scheme. Data of this type are subject to left truncation and right censoring. In many practical cases, two types of censoring may occur. The first type (type A) is due to termination of the follow-up period. The second type (type B) is a consequence of other types of failure which might occur before the cross-section time. Let T*, V*, and denote the lifetime, left truncation, type A and type B censoring variables, respectively. Assume that T*, and are independent of one another but V* and are dependent with . Let F, G and Q denote the common distribution functions of T*, V* and , respectively. Let . For left-truncated and right-censored (LTRC) data, one can observe nothing if Z*<V*, and observe (X*, δ*), if Z*≥V*, where , and δ* is equal to 1 if X*=T*, equal to 2 if and zero otherwise. For LTRC data, the truncation product-limit estimate [Fcirc] n is the maximum likelihood estimate (MLE) for nonparametric models. If the distribution of V* is parameterized as G (x; θ) and the distributions of T* and are left unspecified, the product-limit estimate [Fcirc] n is not the MLE for this semiparametric model. When (i.e., left-truncated data), Wang [Wang M.-C. 1989, A semiparametric model for randomly truncated data. Journal of the American Statistical Association, 86, 130–143.] derived the MLE of F for the semiparametric model and established its weak convergence properties. When G(x; θ)=x/θ and (the so-called stationarity assumption), Asgharian et al. [Asgharian, M., M’Lan, C.E. and Wolfson, D.B., 2002, Length-biased sampling with right censoring: an unconditional approach. Journal of the American Statistical Association, 97(457), 201–209; Asgharian, M. and Wolfson, D.B., 2005, Asymptotic behavior of the unconditional NPMLE of the length-biased survivor function from right censored prevalent cohort data. Annals of Statistics, 33, 2109–2131.] derived an unconditional MLE of F and established its asymptotic properties. In this paper, we extend previous models by distinguishing two types of censoring. Iterative algorithms are proposed to obtain a semiparametric estimate, [Fcirc] n (x; ˆθ n ). The asymptotic properties of [Fcirc] n (x; ˆθ n ) are discussed. A simulation study is conducted to compare the performance of [Fcirc] n (x; ˆθ n ) against that of [Fcirc] n (x).