近二十年来, 越来越多的实证研究表明持久性变化有规律地发生在观测数据中，尤其是在经济, 气候, 水文和金融等方面. 另一方面, 研究者发现大量观测数据难以通过高斯分布描述其统计规律, 而是呈现出尖峰厚尾的特征. 因此, 检验厚尾序列是否存在持久性变点对有效建立模型和准确预测具有重要意义.

       现有的研究厚尾序列持久性变点检验的统计量主要采用最小二乘估计方法, 但该方法对于序列中的异常值非常敏感, 会导致检验功效随着异常值的增多而降低. 鉴于M估计具有良好的稳健性, 本文构造了基于M估计的检验统计量. 具体内容如下:

       在现有的比值型统计量的基础上, 重新提出了基于M估计的比值型统计量. 当持久性变化方向已知时, 推导了在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统计量的一致性. 考虑到原假设下渐近分布形式的复杂性, 采用Bootstrap抽样方法以获取精确的临界值. 数值模拟表明, 基于M估计的比值型检验统计量具备稳健性, 没有出现显著的扭曲, 且相较于基于最小二乘估计的检验统计量明显的提高了经验势. 最后通过上证综合指数, 人民币与美元的汇率两组实证数据进一步验证了文中所给方法的有效性和可行性.

       针对通常情形下持久性变点方向是未知的, 在比值型统计量的基础上, 提出了基于M估计的双侧检验统计量. 基于广义的中心极限定理, 推导了双侧检验统计量在原假设下的渐近分布仍是布朗运动的泛函, 证明了统计量在备择假设下的一致性. 修正的基于M估计的比值型统计量不仅可以根据检验结果判断出持久性变化方向, 且具有良好的经验水平. 即使在异常值较多的情形下, 该统计量仍具备较高的经验势. 最后通过黄金ETF波动率指数和上证综合指数两组实证数据也说明了基于M估计的双侧检验的有效性和可行性.

      During the past two decades, a growing body of empirical research has shown that persistence changes occur regularly in observed data, especially in economic, climatic, hydrology and finance aspects. On the other hand, the researchers found that a large number of observational data could not be described by Gaussian distribution, but showed the characteristics of sharp peaks and heavy tails. Therefore, it is of great significance to test the existence of persistence change points in heavy-tailed sequences for effective modeling and accurate forecasting.

      The statistics of persistence change point test of heavy-tailed sequences are mainly based on least square(LS) estimate, but this method is very sensitive to the outliers in the sequence, which will lead to the efficiency of test will decrease with the increase of outliers. In view of the robustness of M-estimates, a test statistic based on M-estimates is constructed in this paper. The details are as follows:

     On the basis of the existing ratio statistics, a new ratio statistic based on M-estimates is proposed. When the direction of persistence change is known, it is deduced that the asymptotic distribution of the statistics is a functional of Brownian motion under the null hypothesis, and the consistency of the statistics under the alternative hypothesis is obtained. Considering the complexity of the asymptotic distribution under the null hypothesis, the Bootstrap sampling method is used to obtain the accurate critical values. The numerical simulation results show that the ratio test statistics based on M-estimates are robust, without significant distortion, and significantly improve the empirical potential compared with the test statistics based on LS estimate. Finally, two sets of empirical data of the Shanghai Composite Index and the exchange rate between RMB and US dollar are used to further verify the effectiveness and feasibility of the proposed method.

      Since the direction of persistence change point is unknown in common cases, a two-sided test statistic based on M-estimates is proposed on the basis of ratio statistics. Based on the generalized central limit theorem, it is deduced that the asymptotic distribution of the two-sided test statistics under the null hypothesis is still a functional of Brownian motion, and the consistency of the statistics under the alternative hypothesis is proved. The revised ratio statistics based on M-estimates can not only judge the direction of persistence change according to the test results, but also have a satisfactory empirical size. Even in the case of many outliers, this statistic still has a high empirical power. Finally, two sets of empirical data of gold ETF volatility index and the Shanghai Composite Index also demonstrate the validity and feasibility of the two-side test based on M-estimates.