Analysis Of Variable Point And Nonparametric Regression Model

A large amount of finance and meteorology datasets have "heavy tail" characteristic. The tail of the data contains abundant information. Heavy-tailed sequence can well describe this information. Thus many re-searchers pay close attention to this field. Because nonparametric regression model has little limited condition, it has steady result and be extensively used. This paper studies the change point problem of heavy-tailed sequence and nonparametric regression model. The contents of the thesis are as follows:(1) We consider the problem of a mean change in heavy-tailed observa-tions. Firstly, a nonparametric method is proposed to detect mean change. The test statistic is constructed. Under the null hypothesis, the asymptotic distribution of test statistic is obtained. Under the alternative, the consisten-cy of the test is proved. Secondly, a bootstrap method based on ratio test is proposed to detect mean change with heavy-tailed observations. The null distribution of the monitoring statistic and its consistency under alternative hypothesis are proved. To solve the problem that the null distribution of the test statistic contains unknown tail index, we present a bootstrap approxi-mation method to determine the critical values of the null distribution. We prove that the empirical distribution of bootstrap statistic converges in prob-ability to a distribution. And this distribution happens to be the distribution of the test statistic. The theorem not only works in the no-change situation but also establishes in the situation with change point. Thus the consisten-cy of bootstrap test is proved. Then the simulation results are better than those of another existing method. Finally, the empirical application shows the effectiveness of this method.(2) We study the persistence change analysis of heavy-tailed sequence. Firstly, we present new theorem and the corresponding proof about unit root process of the heavy-tailed sequence. Secondly, we discuss the I(0)-to-I(1) persistence change detecting problem. The distribution of the monitoring statistic and its consistency are obtained. Since the null distribution of the test statistic contains unknown tail index, we present a bootstrap test and prove its consistency. The simulation and empirical application illustrate this method is effective. Finally, we consider the I(1)-to-I(0) persistence change testing problem. We derive the null distribution of the statistic and its consistency under alternative. We determine the critical values of the null distribution applying bootstrap method. We prove that the empirical distribution of new statistic converges in probability to the distribution of the original statistic. The excellent performance of our method is demonstrated through simulations using artificial and real datasets.(3) We research change point problems of nonparametric regression mod-el. Firstly, we propose a method based on wavelet analysis to detect and es-timate jump points in nonparametric regression function. The test statistics are constructed on the empirical wavelet coefficients. Under the null hypoth-esis, the threshold and critical values of test statistics are obtained. Under the alternative, the consistency of the test is proved. At the same time, we give the estimators of the number and locations of change points theoretical-ly. The rate of convergence for the change point estimators is established. The simulations and empirical application show excellent performance of our method. Secondly, we consider the variance change of this model. We con-struct the CUSUM test statistics and obtain the asymptotic distribution. We improve and develop the work of previous researchers. The simulation shows this method is effective.