M-Procedures For Detection Of Structure Change Point Under Heavy-tailed Dependent Observations

In the modeling of financial data,in order to avoid the unnecessary loss caused by the misestimation of risk during the investment process,it is necessary to test the change point and estimate the sudden change moment.Considering that many economic and financial data have the characteristic of "spike thick tail",the heavy-tailed distribution can well describe this characteristic,which has attracted the attention of many scholars.In this paper,the statistical analysis of structural change point of heavy-tailed sequences is studied based on M-estimation.The specific contents are as follows:Based on the least square estimation,the mean change point test of heavy-tailed sequences is studied.Since the innovation process is heavy-tailed dependent sequence,the Block Bootstrap method is considered to be used for sampling.In the process of testing,it is found that when the change point is located in the second half of the test,the test effect is relatively poor.Therefore,two methods of reversing statistic and reversing sequence are proposed to modify the test statistics,and the proof is given in theory.The numerical simulation results show that the new statistic test based on the method of reversing statistic is slightly affected by the change point location,compared to the original statistics.When change point is located in the first half of the test,the new statistic test has lower power,when change point is located in the middle and the second half of the test,the empirical power of new statistic test increases,and the change point is located in the second half of the test,its power increases more dramatically.Compared with the original statistic,the power of the new statistic based on the reversed sequence are all increased,and when the change point is located in the first half and the middle of the test,the power of the new statistic is greater than that of the statistic based on the reversed statistic.Although the above test statistics can detect the mean change point of the heavy-tailed sequence,they are easily affected by the heavy-tailed index.A more robust test statistic based on sign function is proposed to research the test problem of the position change point of heavy-tailed sequence.Based on the generalized central limit theorem,it is proved that the asymptotic distribution of the statistic is a functional of Brownian motion under the null hypothesis,and the consistency of the test is obtained under the alternative hypothesis.The numerical simulation results show that the test statistics based on the sign function are more robust and can be effectively tested for the heavy-tailed index ??(0,2).When ??(1,2),the test effect of the statistics is basically not affected by the heavy-tailed index.Moreover,the power of the test statistic is significantly improved compared with the power of the test statistic based on the least square estimation.Therefore,the test statistic based on the sign function realizes the robust test of the position change point of the heavy-tailed sequence.