Offline Test And Online Monitoring Of Mean Change Point Of Heavy-Tailed Sequence

Statistical inference of change point problems plays a very important role in analyzing data and mathematical modeling.In practical applications,the Gaussian sequence cannot describe the characteristics of high-frequency data with sharp peaks and heavy tails,and the theoretical study of the Gaussian sequence change point problem has been very perfect.On the other hand,there are many outliers in the heavy-tailed sequence,showing the characteristics of sharp peaks and heavy tails,which is difficult to test the change point,and the theoretical research is not perfect.Therefore,the change point test of heavy-tailed sequence has become one of the hot research contents that need to be overcome in the current change point problem.In addition,there are both offline data and online data in real life,and it is not feasible to use offline testing for online data.Therefore,this paper will conduct offline testing and online monitoring of the mean change point of the heavytailed sequence respectively,as follows:The content of the offline test is mainly based on the method of least squares estimation,and the problem of the mean change point test of the heavy-tailed sequence of the p-order autoregressive process is discussed.In order to obtain the exact limiting distribution of the test statistics and eliminate the influence of the position of the change point on the test,two modified Ratio test statistics are proposed:the supremum test statistic and the integral test statistic.Secondly,the limiting distributions of the two test statistics are proved to be functional of Lévy process under the null hypothesis,and their consistency is proved under the alternative hypothesis.In order to avoid the estimation of unknown parameters,Bootstrap sub-sampling method is used to determine the more accurate critical value of statistics,and the consistency of Bootstrap sampling method under the null hypothesis is proved theoretically.Monte Carlo simulations show that the Ratio test statistics based on the Bootstrap method not only controls the empirical size well,but also achieves a satisfactor effect of the empirical power,which greatly improves the test empirical power when the change point is located in the second half of the sample.In addition,the empirical size of the integral-type test statistic under the null hypothesis is closer to the significance level,and the empirical power of the supremum-type test statistic under the alternative hypothesis is higher.Then the conclusions of the two test statistics are extended to the case of two change points,and the specific limiting distribution is given.Finally,the validity and feasibility of the proposed method mean change point offline test are verified by two sets of actual data.The main content of the online monitoring is to consider the online monitoring of the mean change point of the independent heavy-tailed sequence with a heavy-tailed index from 0 to 2 based on M estimation.Based on the modified monitoring statistic,the limiting distribution of the modified monitoring statistic under the null hypothesis is no longer the functional of the Levy process,but the functional of the Brownian motion,and the consistency under the alternative hypothesis is deduced.Numerical simulation shows that the online monitoring of mean change point of heavy-tailed sequence based on M-estimation has a very considerable empirical power and a short average run length,which verifies the rationality and effectiveness of the monitoring method proposed in this paper.