在统计学中, 已有的结构变点的检验主要采用最小二乘估计方法, 然而观测序列常呈现 "尖峰厚尾" 特征, 致使检验不稳健, 因此, 针对重尾序列结构变点的检验问题, 选择更稳健的检验方法成为热点研究内容之一. 基于此, 本文基于M估计残差构造比值型统计量检测强混合重尾序列结构变点, 具体内容如下:

针对因重尾性引起的长期方差最小二乘估计偏差, 构建基于M估计残差的比值型检验统计量. 在一定的约束条件下, 证明了原假设下统计量的渐近分布是布朗运动的泛函, 并得到了备择假设下的一致性. 同时, 本文给出了变点位置的估计量, 并证明了其对真实变点时刻位置估计的相合性. 鉴于由序列相依性导致的经验水平扭曲问题, 采用块自举抽样方法逼近极限分布以获得更为精确的临界值. 数值模拟结果显示: 相较于最小二乘估计, 基于M估计残差的比值型统计量的检验功效有明显提升. 特别是尾部较重的情形下, 该检验统计量也有良好的经验势.

针对结构变点在线监测问题, 重新构造了基于M估计残差的滑动比值型在线监测统计量. 基于广义中心极限定理, 证明了原假设下极限分布仍然是布朗运动的泛函, 并证明了备择假设下的一致性. 数值模拟结果表明: 基于M估计残差的滑动比值型统计量对重尾序列结构变点监测具有较强的鲁棒性, 不仅控制经验水平在显著性附近波动, 也有令人满意的经验势. 同时, 相较于最小二乘估计, 基于M估计残差的滑动比值型在线监测方法所需更少的平均运行时间. 这验证了本文所构建的M估计残差滑动比值型监测统计量可行性和合理性.

最后, 选取了美国铝业股票及美元对人民币汇率两组实际数据分别进行线下检验和在线监测. 结果显示文中给出的检验方法能快速有效检测出结构变点, 这表明研究方法是对重尾序列结构变点检测的有效工具.

In statistics, the least squares estimation method is mainly used for the test of existing structural change points, but the observation sequence often shows the characteristics of "spike and thick tail", which makes the test unstable, therefore, for the test problem of structural change point of heavy-tailed sequence, the selection of more robust test method has become one of the hot research contents. Based on this, this article constructs a ratio-type test statistic based on M estimation residuals to detect the structural change points of strongly mixed heavy tail sequences, as follows:

Aiming at the long-term least squares estimation deviation of variance caused by heavy tailing, a ratio-type test statistic based on M estimated residual is constructed. Under certain constraints, it is not only proved that the asymptotic distribution of statistics under the null hypothesis is a function of Brownian motion but also obtained the consistency under the alternative hypothesis. At the same time, the estimator of the change point position is given, and the consistency of the estimation of the real change point position is proved. In view of the empirical level distortion caused by sequence dependence, the block bootstrap sampling method is used to approximate the limit distribution to obtain a more accurate critical value. The numerical simulation results show that compared with the least squares estimation, the empirical powers of the ratio statistic based on M estimated residual is significantly improved. Especially in the case of heavy-tailed, the test statistic also has a good empirical powers.

Aiming at the problem of online monitoring of structural change points, the sliding ratio type online monitoring statistics based on M estimated residuals are reconstructed. Based on the generalized central limit theorem, it is proved that the limit distribution under the null hypothesis is still a functional of Brownian motion, and the consistency under the alternative hypothesis is proved. The numerical simulation results show that the moving ratio statistic based on M-estimation residuals has strong robustness for the monitoring of structural change points of heavy-tailed sequences, which not only controls the fluctuation of empirical level near significance, but also has satisfactory empirical power. At the same time, compared with the least squares estimation, the moving ratio-type online monitoring method based on M-estimation residuals requires less average running time. This verifies the feasibility and rationality of the M-estimation residual moving ratio monitoring statistic constructed in this paper.

Finally, two sets of actual data, Alcoa's stock price, and the US dollar/RMB exchange rate, were selected for offline testing and online monitoring, respectively. The results show that the test method presented in this paper can quickly and effectively detect structural change points, which indicates that the research method is an effective tool for detecting structural change points of heavy-tailed sequences.