Analysis Of Change-point For Two Kinds Of The Heavy-tailed Dependent Observations

Heavy-tailed dependent random variables have been increasingly popular in financial time series since they can describe the "heavy tail" of financial data. Especially, ARCH (Autoregressive conditional heteroscadastic) model, which is proposed by Engle who was awarded Nobel Price of economy, have brought the financial time series into a new epoch.This thesis considers the problem of change point in two classes of heavytailed dependent observations, where one class is ARCH and GARCH (Generalized autoregressive conditional heteroscadastic) observations, the other is dependent observation with infinite variance. The contents of the thesis are as follows:(1) Detection of change-point in ARCH and GARCH Models. We studies the mean change point in the process with ARCH innovation by CUSUM(Cumulative sum) statistic. Consistency of change-point estimator is proved and its rate of convergence is established under the weak assumptions. We also proposed RCUSUM (residual cumulative sum) statistic, which can offset the drawbacks, such as low powers of the SCUSUM(squares cumulative sum) statistic, to test parameters change in GARCH models. Under null hypothesis the limiting distribution of the RCUSUM statistic is considered as well.(2) Test for multiple change-point in ARCH models. We propose a new pseudo-likelihood-ratio test which is based on the CUSUM statistic. The limiting distribution of the test under null hypothesis is present. In addition, We derive analytical expression for asymptotic critical value. The consistent estimator of the location and the number of breaks is instantaneously obtained via the recursive test procedure. The results of simulation study and real data analysis support the validity of methods.(3) We study the problem of a mean change-point in heavy-tailed dependent observations with infinite variance. We get the consistent and rate of change point estimator by CUSUM statistic. To improve the convergence rate, a method of change-point estimation by truncating initial process is proposed as well. In the infinite variance case, we obtained a generalized Hejek-Renyi type inequality. Consistency and the improved rate of convergence for the estimated change-point are also established.(4) Test for mean change-point of heavy-tailed observations with infinite variance. We propose the Subsampling method to detect changes and show that the empirical distribution of subsampling test converges in probability to that of CUSUM test under null hypothesis. Simulation study and real data analysis support the validity of methods(5) Detection structure change in persistence with infinite innovations. We propose the subsampling method to detect changes from I(1) to I(0) and Bootstrap method for changes from I(0) to I(1). We als0 establish the asymptotic validity of these methods. Simulation study assesses the performance of the tests in finite samples.(6) Test for jump points in nonparametric regression function with infinite innovations by wavelet method. The critical value of test is obtained, and the consistency of the wavelet tests is established. Under the alternative hypothesis, we also obtained the consistent estimation of the number and location of the change points. Examples of thresholding empirical wavelet coefficients to test change points are presented.