Asymptotic Theory For The Relative-Risk Model With Missing Time-dependent Covariate

The relationship between a time-dependent covariate and survival times is usually evalu-ated via the Relative-Risk Model and the Additive Model. Time-dependent covariates are gen-erally available as longitudinal data collected regularly during the course of the study. Underthis situation, some papers have given the estimator and the asymptotic theory of the parameterwhich we interested in. A frequent problem, however, is the occurence of missing covariatedata.Recently, an approach to deal with missing time-dependent covariate data is to joint modelswith survival and the longitudinal covariate. However, theoretical justification of this approachis still lacking. In this paper, we first consider the Relative-Risk model with Nonmissing atRandom covariates, and build the joint likelihood. Then we prove existence and consistency ofthe nonparametric maximum likelihood estimate for two mixture model belonging to Relative-Risk model. After that, we give the asymptotic normality of the estimators with a consistentestimator of the asymptotic variance for a important additive model belonging to Relative-Riskmodel, and then we brie?y prove the asymptotic normality of the two mixture model.In the last part of this paper, we consider another important model—Additive model, withmissing covariate. We build the joint model and propose estimates. Then we give the samplingmethod and the recursive procedure of EM algorithm under specific circumstances and soassess the estimation theory using numerical simulation.