Ratio Test To Detect Change Points In Heavy-tailed Dependent Observations

In the financial market, some unexpected situations often make it a dramatic change from an unknown moment. In order to avoid the risk of error estimation in the process of investment and produce unnecessary losses, it is necessary to test and estimate the abrupt-change time. Research has shown that the distribution of the financial asset return has the characteristics of "peak" and "heavy tail", and financial time series is dependent, which makes the study of the change-point problem of the heavy tailed dependent sequence attract more and more scholars’ attention. The most common method to test the change point problem of heavy tail sequence is the cumulative sum(CUSUM)method. In order to obtain the asymptotic properties of the test statistic, the existing CUSUM test requires that the estimation of the scale parameter of the model should be consistent under the null hypothesis and alternative hypothesis. In fact, it is not easy to estimate the scale parameters even if the observations are independent, the case of dependent sequences is more complex. In this paper, in order to avoid these difficulties in CUSUM method, the Ratio statistic based on the ratio of residual CUSUM function is constructed to study the problem of change point of the heavy tailed dependent sequence.In this paper, the research on the heavy tailed dependent sequence is mainly divided into two categories: one is stationary GARCH process, the other is heavy tailed dependent random sequences with infinite variance. The research contents of the paper are as follows:Firstly, detection on the single change point of the stationary GARCH process is studied. It is proved that the limit distribution of the test statistic based on the residual cumulative sum is a functional of Wiener process under the null hypothesis. The validity and applicability of the method are verified by numerical simulation and real data analysis.Secondly, the single change point test on the mean of heavy tailed dependent sequences with infinite variance is studied. Under the appropriate assumptions, the limit distribution of the test statistic under the null hypothesis and the consistency of the test are proved. The results from both numerical simulation and real data analysis support the argument.Finally, the multiple change point test of the mean value in heavy tailed observation with infinite variance is studied. The ANOVA-type test statistic is constructed to test theexistence of multiple change points in the sequence, the limit distribution of the test statistic under the null hypothesis is obtained and the consistency of the test under the alternative hypothesis is proved. The availability of the method is verified by numerical simulation.