变点问题的统计推断对分析数据和数学建模有着非常重要的地位。在实际应用中，高斯序列不能刻画呈现尖峰厚尾的高频数据的特性，且高斯序列变点问题的理论研究已经非常完善。另一方面，厚尾序列中存在较多的异常值，呈现尖峰厚尾的特征，对其进行变点检验存在一定困难，且理论研究尚不完善。因此对厚尾序列的变点检验成为了目前变点问题需要攻克的热点研究内容之一。此外，在现实生活中同时存在着线下数据和在线数据，对于在线数据利用线下检验的方法并不可行，因此本文将对厚尾序列的均值变点分别进行线下检验和在线监测，具体如下：

       线下检验的内容主要基于最小二乘估计的方法讨论了p阶自回归过程的厚尾序列均值变点检验问题。为了得到检验统计量精确的极限分布和消除变点位置对检验的影响，提出了两个修正的Ratio检验统计量：supremum型检验统计量和integral型检验统计量。其次在原假设下证明了这两种检验统计量的极限分布是Lévy过程的泛函，并在备择假设下证明其一致性。为了避免对未知参数的估计，采用Bootstrap子抽样方法确定更精确的统计量临界值，并理论证明了原假设下Bootstrap抽样方法的一致性。蒙特卡洛模拟表明，基于Bootstrap方法的Ratio检验统计量不仅很好的控制了经验水平，且经验势也达到令人满意的效果，大幅度提高了变点位置位于样本后半段时检验的经验势。此外在原假设下integral型检验统计量的经验水平更接近于显著性水平，备择假设下supremum型检验统计量的经验势更高。随后将两个检验统计量的结论推广到了两个变点的情形，并给出了具体的极限分布。最后通过两组实际数据验证了所提均值变点线下检验方法的有效性和可行性。

       鉴于基于最小二乘估计的厚尾序列均值变点检验严重受厚尾指数的影响，在线监测的主要内容考虑了基于M估计的厚尾指数为0到2的独立厚尾序列的均值变点在线监测。基于修正的监测统计量，得到原假设下修正的监测统计量的极限分布不再是Lévy过程的泛函，而是布朗运动的泛函，并推导了备择假设下的一致性。数值模拟表明了基于M估计的厚尾序列均值变点在线监测的检验势达到非常可观的程度，且平均运行长度也较短。这验证了本文所提监测方法的合理性和有效性。

      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 pe- aks 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 con- duct offline testing and online monitoring of the mean change point of the heavy- tailed 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 seq- uence 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 altern- ative hypothesis.In order to avoid the estimation of unknown parameters,Bootstrap sub-sampling method is used to determine the more accurate critical value of stati- stics, 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 monitori- ng 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 Lévy 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.

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