Almost sure representations in survival analysis under censoring and truncation. Applications to goodness-of-fit tests

Abstract

This chapter presents a generalized product-limit estimator of the conditional distribution function when the data are subject to random left truncation and right censorship (LTRC). This result extends strong representations studied on conditional survival analysis for censored data as well as for truncated data. A PLE for LTRC data when covariables are present is considered with an almost sure asymptotic representation for it. The use of smooth nonparametric estimators for the conditional distribution functions is emphasized. One of the main aims of the survival analysis, in a conditional setup, is to analyze how an explanatory variable influences the survival time. This dependency may be modeled in a different number of ways but, the key idea is to assume some kind of functional relationship for the conditional distribution function or some other conditional curve. The idea behind proportional hazards model is to split the conditional hazard rate as a product of two factors, which include one independent of the covariates and another covariate dependent. The test statistics have to be multiplied by a normalizing sequence in order to have a limit distribution.