On The Statistical Inference On Change-point Problem In Segmented Linear Regression Models

Since 1950s, the detection and estimation of change-point has been one of the hot topics in statistics. The reasons that we are interested in it are that it has not only important application to industrial quality control, but also can be encountered in many other fields such as economics, finance, medicine, epi-demiology, psychology, environment science, meteorology, and geology. From the statistical point of view, a change point refers to an observational time point at which a statistical pattern change occurs during a long-term process or sequence. The research about change-point mainly focus on two aspects:one is to decide if there is any change (often viewed as an hypothesis testing problem), another is to locate the change-point when there is a change present (often viewed as an estimation problem). In this dissertation, we mainly discuss the change-point problem in segmented linear regression models. This dissertation has five chapters that are organized as follows:In Chapter 1, we introduce the backgrounds and significance of change-point problem and give two models about it, namely, the change-point in distribution functions and the change-point in regression models.The existing methods used to study the change-point problem are summa-rized in Chapter 2. It includes parameter likelihood ratio method, the test based on recursive regression residuals and regression residuals, union-intersection test and the method of information criterion.In Chapter 3, we describe the empirical likelihood method introduced by Owen (1988). Empirical likelihood method combines the reliability of the non-parametric methods with the flexibility and effectiveness of the likelihood ap-proach, so it has many outstanding advantages, compared to other classical meth-ods. This chapter generalizes the empirical likelihood inference about statistical functional and regression model.In Chapter 4, we study the change-point problem in segmented linear re-gression models via the empirical likelihood method. We study the large sample properties of the ad hoc empirical likelihood detection procedure proposed by Liu and Qian (2010) and derive the limiting null distribution of the corresponding test statistic. The limiting null distribution appears to be the same as that of likelihood ratio test statistics given by Csorgo and Horvath (1997). In the course of development of the large sample theory, it is proved that the Lagrangc multi- plicr in the empirical likelihood method setting under the segmented regression models has similar large sample property as in the ordinary cases studied by Owen (1990). This result can greatly help us understand the ad hoc empirical likelihood detection procedure better and is very useful in numerical approximation to the Lagrange multiplier that is critical in use of empirical likelihood methods. In this chapter, we also propose a new detection procedure to improve the efficiency of the ad hoc empirical likelihood detection procedure. The main idea behind the new detection procedure is to transform the highly correlated residuals on which the ad hoc empirical likelihood method is based to uncorrelated quantities and then the norm equation of regression analysis based on the transformed responses is used as the constraints of the empirical likelihood method. The new detection procedure is computationally as easy as the ad hoc empirical likelihood detection procedure and is expected to be a great improvement. Monte Carlo simulation studies are conducted. The simulation results show that the new detection pro-cedure is indeed substantially more efficient with finite samples. Furthermore, the null limiting distribution of the new detecting test statistic is remarked and a bootstrap estimation procedure is proposed to approximate the new test p-value for the purpose of applications in practice.In Chapter 5, Liu and Qian (2010)'s detection procedure and the new detec-tion procedure we propose are applied to a real-life data set for the eruptions of the Old Faithful geyser in Yellowstone National Park in U.S.A. The same data set was analyzed by Gombay and Horvath (1994). We reanalyze the relationship between the duration time and the interval about the eruptions of the Old Faith-ful geyser. The scatter plot of 270 eruptions in October 1980 clearly suggests that the geyser eruption has a change point during October 1980. It is indeed turned out that the p-value based on the new detection procedure is 0.05 and the p-value based on Liu and Qian (2010)'s procedure is 0.07, in sharp contrast to the p-value 0.17 based Gombay and Horvath (1994) under normal parametric models.In the last chapter, we give a summary of the dissertation and outline a future research plan.