Some Topics On Change Point Problem In Quantile Regression

The change point problem is one of the hot topics in statistical research since 1970s. Change point detection is not only widely applied to the industrial field of quality control (the earliest change point application field), but also used in economics, finance, medical science, computer science and other fields. In detail, the main performances and results of this study are described as follows:(1). We develop a new procedure for testing change points due to a covariate threshold in threshold quantile regression model. The proposed test is based on the CUSUM of the subgradient of the quantile objective function and requires fitting the model only under the null hypothesis. The critical values can be obtained by simulating the Gaussian process that characterizes the limiting distribution of the test statistic. The proposed method not only can be used to detect change points at a single quantile level or across multiple quantiles, but also can accommodate both homoscedastic and heteroscedastic errors. The efficiency and feasibility of the proposed method is confirmed by some simulation studies and a real data application.(2). We propose a method to estimate the change point in bent line regression model. In some applications, the change points at which the quantile functions are bent tend to be the same across quantile levels or for quantile levels lying in a certain region. To capture such commonality, we propose a composite estima-tion procedure to estimate model parameters and the common change point by combining information across quantiles. We establish the asymptotic properties of the proposed estimator, and demonstrate the efficiency gain of the composite change point estimator over that obtained at a single quantile through numerical studies. In addition, three different inference procedures including Wald-type test, bootstrap and rank score test are proposed and compared for hypothesis testing and the construction of confidence intervals. Some simulation studies and a real data application suggest that the proposed estimator has a stable and competitive performance.(3). We develop a new method to test change points in censored quantile regression with panel data. The testing procedure is based on the observations in an informative subset and conducted with the subgradient of the quantile objective function, which only requires estimating the model under the null hypothesis. The proposed method is easy to understand and convenient to compute. We establish the limiting distribution of the test statistic under the null hypothesis, and shows that its asymptotic critical values can be obtained via simulation method. The finite sample performance of the proposed method is examined by some simulation studies and a real data analysis.(4). Based on local linear regression, we provide a method to test the existence of the change points in partially linear models. We propose the test statistic and establish its asymptotic properties under the null and local alternative hypotheses. The finite performances of the proposed method are illustrated by some simulation studies and a real data analysis.The innovations of the achievements in this dissertation are described as fol-lows. Firstly, we consider change point problems in quantile regression models, which enrich the study of the change point. Secondly, a simple and efficient testing procedure is provided for testing for change points due to a covariate threshold in threshold quantile regression model. Thirdly, based on composite quantile re-gression, we provide a more efficient estimation method for the change points in bent line regression model. Fourthly, we propose a method to detect the existence of the change points in censored quantile regression with panel data. At last, we consider a simple procedure to test the existence of the change points in partially linear models.The innovations of the methodologies in this dissertation are described as follows. Firstly, based on Score-type testing procedure, we first provide a simple and robust method for detecting the change point in threshold quantile regression model. Secondly, making use of the information jointing modeling of multiple quantiles, the proposed change point estimation in bent line regression model is rather efficient. Thirdly, a simple and effective method for testing the change point in censored quantile regression with panel data is provided. At last, based on local linear regression, we propose a simple testing procedure for the jump in the nonparametric function in partially linear model.The achievements and methodologies in this dissertation enrich the theory of change point problems, which also help to solve the problem of change point detection and estimation in quantile regression, and promote the application of change point in many fields, such as quality control, economics, finance, medical science, computer science and other fields.