The Application Of Martingale Analysis In Survival Analysis Models

Survival analysis, which is an important branch of mathematical statistics, has developed very quickly since 1970s. It originated from lots of practical problems in modern medicine and engineering, stresses statistical analysis of censored data, has great practicability and contributes a lot to reliability statistics of products in medicine and engineering. By applying some latest theory of probability theory and statistics, survival analysis not only deals with censored data problem in real life, but also reveals more complicatedly theoretic problems and promotes the development of mathematical statistics, while it settles down practical problems.First, in this paper, based on the fundamental theory of survival analysis and martingale, the partial maximum likelihood estimate method is used to confirm the form of parametric estimators for the Cox proportional hazards model for censored data with time-dependent covariates which have a stratum effect on the hazard function of the life-time distribution of an individual, and the martingale analysis method is taken to gain the estimators are consistent and asymptotic normal. This shows that in clinical trials using partial maximum likelihood method to estimate the parameters of the model is feasible. Then two methods as parametric test and nonparametric test are used to hypothesis testing each baseline hazard function, and two corresponding test statistics are constructed by combined to martingale central limit theorem.Second, martingale analysis method is introduced into the Cox proportional hazards model for censored data with time-dependent regression coefficients, the locally linear partial maximum likelihood estimate method is used to confirm the form of parametric estimators, and the martingale analysis method is applied to obtain that both the locally linear partial maximum likelihood estimators for the time-dependent regression coefficients and the nonparametric estimator for the baseline cumulate hazard function is consistent. This shows that bootstrapping the model is feasible. Therefore, a reliable theoretical basis is provided for the actual application of survival analysis models.