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-Perez and Gonzalez-Manteiga (1999), 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), a least squares estimator of β(y) is obtained and a bootstrap procedure is proposed to choose the optimum bandwidth.