Statistical Inference In Change-point And Its Application In Finance

Change-point problems have originally arisen in the context of quality control, where one typically observes the output of a production line and would wish to signal deviation from an acceptable level while observing the data. It also occurs in many natural and social fields such as economic, finance, seismology, epidemiology and computer science. In this thesis, we discuss mainly the statistical inference of change-point problems inΓ-distribution and its application in finance.This thesis is divided into two parts. In the first part, including the first four chapters, the statistical inference of change-point problems are mainly considered. To apply the theory and methods of change-point problems in practice, we discuss the application of change-point problems in finace in the second part (Chapter five).The development and research status of change-point problems are briefly introduced in the first chapter.Former research focused on change-point problems in mean and variance mostly, where the mean and variance are independent parameters. But the mean and variance ofΓ-distribution depended on common parameters v and A simultaneously. It is an interested topic that how to detect the change-point in distribution with multi-parameter which are interrelated. In the second chapter, the change-point of parameters inΓ-distribution is considered. Suppose that are independent random variables, such that is unknown and called change-point. With help of the theory of Gaussian process and slipping window, the asymptotic distribution of the statistic proposed in this chapter can be approximated by the first type of extremal distribution. Hypothesis test and estimation about change-point are considered. Furthermore, the consistency and convergence rate of change-point are also presented. At the same time, strong rate and weak rate of convegence of change-point are established. Lastly, simulations by Matlab are displayed, it is shown that our procedures are rather accurate.In chapter three, the procedures about detectection and estimation of change-point are proposed with the help of CUSUM. In the mild conditions, the strong consistency and convergence rate of estimator of change-point are also presented. The estimation of the magnitude of jump v^1/2 = v₂- v₁ and the asymptotic distribution of estimator are also discussed. At the same time, the asympotic distribution of estimator of change point is also presented when the variance of X_i is known. The procedures to detect the change-point are also proposed by mean of self-normalization when the variance of X_i is unknown. At last, the results of simulation is presented, it is shown that our test and estimator are rather efficient.In the fourth chapter, we discussed the change-point problem with at most one change inthe model: where are independent and identically distributed. The strong and weak consistency of estimator of change-point are considered. The strong and weak convergence rate of estimatorÏ„|^ of change-point are also given. In the mild conditions, the asymptotic distribution of estimatorÏ„|^ is presented.As a practical application of change-point problem, in the fifth chapter we research the successive rises and falls of Shanghai stock returns by change-point analysis methods inΓ-distribution. Based on the the example of Shanghai stock index returns, the following result is gained: The distributions of the successive rises and falls of returns are fitted with Gamma- distribution well in period " T+0 ", " T+1" " rise and stop " " reduce state stake in listed companies " respectively, but the shape parameter and scale parameter inΓ-distribution are not the same in different periods. This reflects the trait of " policy " on the course of development in Chinese stock market.