Empirical Likelihood Inference Of Some Change-Point Models

The problem of change-point has been a hot problem of permanent interest in statistical literature, it is widely used in inustrial quality control, image processing, computer, economic, signal processing, medical and other fields. From the statistical point of view, the so-called change-point refers to an observational time point at which a sequence or process changes occurs, this change will generally reflect certain qualitative sequence or process changes. The research about change-point problems mainly focus on two aspects: one is to detect if there is any change, that is often viewed as a hypothesis testing problem. Another is to make a change-point statistical inference when there is a change-point, that is often viewed as an estimation problem. The research methods of change-point problem mainly include three categories: parametric methods, bayesian methods and nonparametric methods. Parametric methods generally require a sample model assumptions, if the assumption of model is an error, the result will be an error, the most common method is maximum likelihood method in parameter methods. Bayesian methods derive posterior distribution about the parameter of change point based on the prior distribution, and make a statistical inference. Nonparametric methods do not do need to make assumptions to model, and obtain good results to a certain extent. Empirical likelihood is a nonparametric method proposed by Owen(1988), and has many good properties. In this paper, we mainly discuss the change-point problem in generalized linear models under natural connection and censored linear regression model based on empirical likelihood method. First, empirical likelihood ratio test statistic and the estimation of change point are conducted by establishing a change-point model. Then, under the conditions of the original assumption, the empirical likelihood ratio test statistic is proved to have the same asymptotic distribution as that with classical parametric likelihood. And further prove the large sample properties of the estimate of change-point. Finally, the simulation results and a real example demonstrate the feasibility of the proposed approach. The main contents are organized as follows:In chapter 1, we first give a brief overview of the research background and common research content of change-point. Then, we introduce the empirical likelhood method. Finally, we describe the the main content and innovation of paper.In chapter 2, we mainly discuss the change-point problem in generalized linear models under natural connection based on empirical likelihood method. First, we give a brief overview of generalized linear models. Then, under the conditions of the original assumption, the empirical likelihood ratio test statistic is proved to have the same asymptotic distribution as that with classical parametric likelihood. And further prove the large sample properties of the estimate of change-point. Finally, simulation is based on logistic regression model the most common example of generalized linear model as simulation object, and a real example demonstrate the feasibility of the proposed approach.In chapter 3, we mainly discuss the chang-point problem in the censored linear regresion model based on the adjusted empirical likelihood method. First, we give a brief overview of censored data. Then, considering the most common censored linear regression model, and building change point model. Since the raw data exists censored, the raw data is transformed into pseudo complete data by the method of Buckly & James(1979). Since the empirical likelihood method generally requires estimation equation variables are independent of each other, we can not get the classic asymptotic distribution by general empirical likelihood. Therefore, we obtain the asymptotic distribution of the adjusted empirical likelihood ratio test statistic by constructing an adjusted factor. Finally, the simulation results and a real example demonstrate the feasibility of the proposed approach.In chapter 4, we give a summary of the dissertation and outline a future research plan.