在对金融数据进行建模时，为了避免在投资过程中对风险错误估计产生不必要的损失，需要对变化情况进行检验并对这个突变时刻进行估计。许多经济和金融数据具有峰态厚尾的特征，厚尾分布能很好地刻画这一特性，引起众多学者的关注。本文基于M-估计对厚尾序列结构变点的统计分析问题进行了研究，具体内容如下：
       基于最小二乘估计构造检验统计量研究厚尾序列的均值变点检验问题，由于新息过程为厚尾相依序列，考虑利用Block Bootstrap方法进行抽样。在检验过程中发现当变点位置位于后半段时，检验效果相对较差，基于此，本文提出利用颠倒统计量和反转序列这两种方法进行修正，并在理论上给出了证明。数值模拟表明：相比于原统计量，基于颠倒统计量构造的新统计量的检验效果受到变点位置的影响较小，当变点位置位于前半段时，经验势函数值稍微降低；当变点位置位于中间和后半段时，经验势函数值增加，且变点位置位于后半段时增加的幅度更大。相比于原统计量，基于反转序列构造的新统计量的经验势函数值均有增加，且变点位置位于前半段和中间时，经验势函数值大于基于颠倒统计量构造的新统计量的经验势函数值。
      上述基于最小二乘估计构造的统计量虽可以检验厚尾序列均值变点，但容易受到厚尾指数的影响，由此本文基于符号函数构造更稳健的检验统计量来研究厚尾序列位置变点的检验问题。基于广义中心极限定理，在原假设下证明统计量的渐近分布是布朗运动的泛函，并在备择假设下得到了检验的一致性。数值模拟表明：基于符号函数构造的检验统计量对厚尾指数 α∈(0,2) 均可以有效检验。当 α∈(1,2) 时，统计量的检验效果基本不会受到厚尾指数的影响，并且检验统计量的经验势函数值相比于基于最小二乘估计构造的检验统计量有明显的提高。因此，本文基于符号函数构造的检验统计量实现了厚尾序列位置变点的稳健检验。
 

       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 α∈(0,2) . When α∈(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.
 

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