A Generalized Time-Dependent Conditional Linear Model with Left-Truncated and Right-Censored Data

Consider the model φ(S(y | X)) = β(y) ^T X, where φ is a known link function, S(· | X) is the survival function of a response Y given a covariate X = (1, X, X ²,…, X ^p ), and β(y) is an unknown vector of time-dependent regression coefficients. The response Y is subject to left truncation and right censoring. We assume that given X, Y is independent of (C, T) where C and T are censoring and truncation variables with P(C ≥ T) = 1. In this article, with some modification of the assumptions in Lemmas 5 and 6 of Iglesias-Pérez and González-Manteiga (1999 Iglesias-Pérez , C. J. , González-Manteiga , W. G. ( 1999 ). Strong representation of a generalized product-limit estimator for truncated and censored data with some application . J. Nonparametric Statist. 10 : 213 – 244 .Taylor & Francis Online], Web of Science ®] , Google Scholar]), we present an almost sure representation for the generalized product-limit estimator (GPL) of S(y | X). Based on the GPL and the approach of Teodorescu et al. (2010 Teodorescu , B. , Keilegom , I. V. , Cao , R. ( 2010 ). Generalized time-dependent conditional linear models under left truncation and right censoring . Ann. Instit. Statist. Math. 62 : 465 – 485 .Crossref], Web of Science ®] , Google Scholar]), a least squares estimator of β(y) is obtained and a bootstrap procedure is proposed to choose the optimum bandwidth.