The Application On Survival Analysis By Martingale

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 and methods of survival analysis, martingale analysis method of point process in stochastic process is introduced and product limit estimator of non-parametric estimation is discussed. Moreover, we introduce survival analysis experiment with right-censored data into strong stationary-weakly mixing process that parameter is continuous and state is continuous, express survival data sample as counting process and introduce martingale analysis method of point process. Furthermore, asymptotic unbiasedness of product limit estimator of survival analysis and weak convergence of product limit estimator in stochastic process are discussed and a method of calculating product limit estimator by martingale method is presented. Besides, we study functional estimators of survival functions in strong stationary-weakly mixing process, including failure fate function, cumulative failure fate function, mean survival time function, residual life distribution function, mean residual life time, and almost sure convergence properties of these estimators.Second, we introduce martingale method into increased reliability experiment of inhomogeneous Poisson process which is with right-censored data, and fix the shape parameters of model with right-censored data by maximum- likelihood estimation. By transforming inhomogeneous Poisson process into temporally homogeneous Poisson process, we study mean time problem in malfunction by martingale analysis and time interval problem in malfunction by stopping time theory of martingale, and obtain some related results.