Block Bootstrap For Detection Of Mean Change Point Under Heavy-tailed Model

Since the 1970s,the change point problem has been one of the hot topics in econometrics.The theory and practical application of change points have attracted widespread attention.The change point directly affects the distribution characteristics of the series.In order to build a more accurate time series model,so it is very important for testing the structural point.Since heavy-tailed series are characterized by "spikes and heavy tails",and the data are often dependent on each other.The mean is also the main numerical characteristic of time series data.Based on the characteristic,this paper uses Block Bootstrap sampling method to detect the mean-change point under heavy-tailed model.The detailed content is as follows.When the heavy-tailed index is(1,2),this paper adopts the least squares estimation method to construct a robust ratio statistic for detecting the mean change points,and the innovation process is AR(p).Under reasonable assumptions and lemmas,the limiting distribution of the test statistic under the null hypothesis and its consistency under the alternative hypothesis are theoretically derived.However,it is found that the test effect of the original Ratio is easily affected by the location of the change point,and the empirical potential of the change point occurring in the first half of the period is higher than that of the change point occurring in the second half of the period.Based on the defect,the statistic was reversed to derive the reversal statistic,and then the maximum of the original statistic and the reversal statistic was taken.Thus theoretically confirming that the empirical potential of the revised statistic is no longer sensitive to the moment of the change point.Since there is a certain degree of inter-series dependence and the critical value of the statistic is related to the heavy-tailed index,the Block Bootstrap sampling method is used to simulate a more accurate critical value.Finally,numerical simulations with limited samples can conclude that the modified statistic can effectively test the mean change point,which also proves the feasibility of the method.When the heavy-tailed index is(0,1),this paper adopts the M-estimation method to study the mean change point of the heavy-tailed series with the innovation process as AR(1).When the heavy-tailed index is small,the number of the singularity becomes more and the series has no expectation,so least squares estimation will be weakened.Thus the M estimation method is proposed.Based on the generalized central limit theorem,the limiting distributions of the statistics under the null hypothesis are derived,and their consistency under the alternative hypothesis is determined.Numerical simulations under finite samples show that the critical values of the statistics for different parameters are relatively stable,the empirical size under the null hypothesis fluctuates around 0.05,and the empirical potential under the alternative hypothesis also increases with the increase of the sample size and the jump magnitude.Therefore,when there are more anomalous data,the M estimation is more robust than the statistic constructed by the least squares method.