Change-point Detection In Survival Models

Since introduced in the 1950s,the change-point problem has always been a hot research field in mathematical statistics.Research on the change point is often concerned with testing whether the change point exists and estimating the number and location of the change points.Most research focuses on the change-point detection in independent Gaussian sequences,and there have been many classic research results with theoretical basis.However,in survival analysis,the algorithm of change-point detection needs to be optimized urgently,and related theories need to be perfected.This dissertation focuses on the change-point detection in survival analysis,and discusses two types of change-point models.One is the piecewise constant hazard function with the change point,which can be applied to detect the time point when the hazard changes suddenly.The estimation of the change point can be obtained by the geometric characteristics of the cumulative hazard function.The other type is the threshold model where the change point exists on the covariate.This model can be applied to detect the heterogeneity of data,which means there are unknown groups in the observations,and the regression coefficient in each group has a different value.The change point is usually estimated by the maximizing the likelihood function.Since the likelihood function is not continuous with respect to the change point,the maximized likelihood estimation of the change point can be obtained by combining the profile-likelihood function and the grid search method.Corresponding to different models and estimation methods,this dissertation establishes the theoretical properties of change-point estimator and provides methods for the change-point testing.For the second type of model,the detection method of multiple change points is also discussed in this dissertation.The content of this article has three parts.The first part studies the change point detection in the piecewise constant hazard function.Consider that there is a cure-fraction in the population,and the data is interval-censored.When estimating parameters,we first apply the non-parametric method to obtain the estimation of the cure rate and the change point,and then utilize the pseudo-likelihood method to estimate the hazards of each segment in the model.We discuss the asymptotic properties of the estimators for the change point and other parameters and apply the modified likelihood ratio statistic to test the existence of the change point.The second part discusses the change-point detection in the Cox mixture cure model,where the change point is located on a continuous covariate.A two-step method combining grid-search and profile likelihood is applied for estimation.By the m out of n Bootstrap and the method of Louis,we obtain the variance estimation of the change-point estimator and regression parameter estimators,respectively.We establish the asymptotic properties of the estimators and discuss the change-point test problem.The first two parts only discuss the detection of the single change point,the third part studies the multiple change-points detection problem under the Cox proportional hazards model.The estimation of the change-points number and locations are obtained by a two-stage method.Specifically,the first stage estimates the number of change points and determines the segments containing the change points by splitting the data and constructing the penalized partial likelihood.The second stage determines the locations of the change points by the single change-point detection.The proof of the consistency for change-points estimator takes into account the theoretical results of the model misspecification and the high-dimensional sparse estimation.In each part,we have done sufficient simulation studies and real data analysis to verify the finite sample performance of the proposed methods.