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

Consider the model phi(S(y vertical bar X)) = beta(y)X-T, where phi is a known link function, S(. vertical bar X) is the survival function of a response Y given a covariate X = (1, X, X-2, ... , X-p), and beta(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-Perez and Gonzalez-Manteiga (1999), we present an almost sure representation for the generalized product-limit estimator (GPL) of S(y vertical bar X). Based on the GPL and the approach of Teodorescu et al. (2010), a least squares estimator of beta(y) is obtained and a bootstrap procedure is proposed to choose the optimum bandwidth.