变点问题一直是统计学中的一个热门话题，主要研究时间序列数据在某个时刻是否存在分布或数字特征的变化，这个时刻就叫变点。对于稳定分布，其尾指数刻画了分布的尾部概率，度量了系统的风险。越来越多的研究表明，尾指数会随着时间的推移而发生变化。若忽略这种变化，可能会造成预测结果不准确，从而导致错误的判断分析。因此，厚尾相依序列尾指数变点检验是亟待解决的问题。主要研究内容如下：

       研究厚尾相依序列尾指数变点的线下检验问题，已有文献一般基于最小二乘估计的残差构造检验统计量，但需限定尾指数属于区间(1,2]，这极大限制了其应用范围。文章采用M估计方法将尾指数拓展为(0,2]，并证明了该方法下参数估计的一致性。在此基础上提出了两类针对尾指数变点的比值型统计量，利用广义的中心极限定理，给出了检验统计量在原假设下的极限分布及备择假设下的一致性。由于极限分布依赖于未知的尾指数，提出Bootstrap抽样方法确定极限分布的临界值。数值模拟结果表明第一类统计量的经验势对样本量大小、变点位置、跳跃幅度等因素较为敏感，而第二类检验统计量受变点位置的影响较小，且相较于第一类统计量具有更好的检验效果。最后通过一组正虹科技股票收盘价格的数据进行了实证分析，进一步验证了文章所采用方法的可行性。

        厚尾相依序列的尾指数变点在线监测问题，针对尾指数变点在线监测问题，为了提升统计量检验的稳健性，构建了基于M估计的滑动比值型监测统计量，证明了该统计量在原假设下的极限分布是维纳过程的泛函，且得到了其在备择假设下的一致性。数值模拟表明原假设下的经验水平值均在0.05的显著性水平附近波动，在备择假设下分析了经验势函数值与变点位置、样本量、跳跃幅度等因素的动态机理，最后通过一组九州药业股票收盘价格的数据进一步验证了文章所提出的监测方法的合理性和有效性。

   The change point problem has always been a hot topic in statistics. It mainly studies whether there is a change in distribution or numerical characteristics of time series data at a certain moment, which is called change point.For the stable distribution, the tail index describes the tail probability of the distribution and measures the risk of the system.A growing body of research shows that the tail index changes over time.If such changes are ignored, the prediction results may be inaccurate, leading to wrong judgment and analysis.Therefore, the test of tail index change point of thick tail dependent sequence is an urgent problem to be solved.The main research contents are as follows:

    The off-line test problem of tail index change point of thick tail dependent sequence is studied. Existing literatures generally construct test statistics based on residual of least square estimation, but the tail index should be restricted to belong to the interval (1,2], which greatly limits its application range.In this paper, the M estimation method is used to extend the tail index to(0,2], and the consistency of parameter estimation under this method is proved.On this basis, two kinds of ratio type statistics aiming at the change point of the tail exponential are proposed. By using the generalized central limit theorem, the limit distribution of the test statistics under the original hypothesis and the consistency under the alternative hypothesis are given.Because the limit distribution depends on unknown tail index, a Bootstrap sampling method is proposed to determine the critical value of the limit distribution.The numerical simulation results show that the empirical potential of the first type of statistic is sensitive to the sample size, change point location, jump amplitude and other factors, while the second type of test statistic is less affected by the change point location, and has a better test effect than the first type of statistic.Finally, through a group of Zhenghong Technology stock closing price data for empirical analysis, further verify the feasibility of the method adopted in this paper.

     The second problem is the online monitoring of the tail index change point of the thick tail dependent series. For the online monitoring of the tail index change point of the thick tail dependent series, the sliding Ratio monitoring statistic is adopted, and the consistency of the limit distribution of the statistic under the original hypothesis and the alternative hypothesis is proved. The numerical simulation shows that the larger the monitoring sample size, the better the test effect, and the empirical level values under the original hypothesis all fluctuate around the significance level of 0.05, and the empirical potential function values under the alternative hypothesis are slightly affected by the change point location, sample size, jump amplitude and other factors, which further verifies the rationality and effectiveness of the monitoring method proposed in this paper.

[1] Page, E. S. Continuous Inspection Schemes[J]. Biometrika, 1954, 42(1): 100-115.

[2] 张梦琇. 基于贝叶斯分析正态分布均值变点的统计推断及应用[D]. 新疆师范大学, 2019.

[3] 雷鸣, 谭常春, 缪柏其. 运用生存分析与变点理论对上证指数的研究[J]. 中国管理科学, 2007, 15(5): 1-8.

[4] 戴迪昊, 钱泽平, 丁明月. 基于Weibull分布的最大风速变点估计[J]. 大学数学, 2018, 34(5): 12-18.

[5] 袁芳. 西安市房地产价格指数波动性分析[J]. 金融经济, 2018(02): 100-101.

[6] Mandebrot BB. The variation of certain speculative prices[J]. Journal of Business. 1992, 36(3): 394-491.

[7] 武东, 汤银才. 稳定分布及其在金融中的应用[J]. 应用概率统计, 2007, 11(3): 434-445.

[8] Ibragimov R. Heavy-Tailedness and Threshold Sex Determination[J]. Statistics and Probability Letters, 2008, 78(12): 2804-2810.

[9] 邵钏. 关于方差变点问题的几个统计推断[D]. 浙江: 浙江大学, 2012.

[10] 谭长春, 赵林城, 缪柏其. 至多一个变点的 分布的统计推断及在金融中的应用[J]. 系统科学与数学, 2007, 27(1): 2-10.

[11] Kim J Y. Detection of Change in Persistence of a Linear Time Series[J]. Journal of Econometrics, 2000, 95(1): 97-116.

[12] Loretan Mico,Phillips Peter C.B.. Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets[J].Journal of Empirical Finance, 1994, 1(2): 211-248.

[13] PAGAN, A.R. and SCHWERT, G.W., Testing for Covariance Stationarity in Stock Market Data[J]. Economics Letters, 1990, 33: 165–170.

[14] Quintos, C., Fan, Z.,& Phillips, P. C. B.. Structural change tests in tail behaviour and the Asian crisis[J]. Review of Economic Studies, 2001, 68: 633–663.

[15] Kim, M., & Lee, S.. Change point test for tail index for dependent data[J]. Metrika, 2011, 74: 297–311.

[16] Kim, M., & Lee, S.. Test for tail index change in stationary time series with Pareto-type marginal distribution[J]. Bernoulli, 2009, 15: 325–356.

[17] Kim, M. & S. Lee Change point test of tail index for autoregressive processes[J]. Journal of the Korean Statistical Society, 2012, 41: 305–312.

[18] Yannick Hoga. CHANGE POINT TESTS FOR THE TAIL INDEX OF β -MIXING RANDOM VARIABLES[J]. Econometric Theory, 2016, 33(4): 915-954.

[19] 胡兴. 基于贝叶斯方法的单参数指数族分布参数变点的统计推断[D]. 新疆大学, 2011.

[20] 彭秋曦. 左截断右删失数据下指数分布变点的Bayes估计[D]. 重庆大学, 2015.

[21] 黄月兰. 单参数指数分布变点估计研究[J]. 广西民族师范学院学报, 2014, 31(03): 1-3.

[22] 王思惟. 基于双参数指数分布参数的统计推断研究[J]. 科技创新导报, 2020, 17(12): 223-225.

[23] 冯娜. 基于滞后系数突变的结构突变AR(p)模型的上证指数变点研究[D]. 华侨大学, 2013.

[24] 周影辉, 倪中新, 朱平芳. 基于最小一乘准则的上证指数突变点研究[J]. 中国管理科学, 2015, 23(10): 38-46.

[25] 梅梦玲, 周菊玲, 董翠玲. IIRCT下指数分布参数多变点的贝叶斯估计[J]. 河南科学, 2021, 39(03): 353-358.

[26] Huber P J. Robust Statistics[M]. New York: Wiley, 1981.

[27] 陈希孺, 赵林城. 线性模型中的M方法[M]. 上海科学技术出版社, 1996.

[28] 白会会. 基于M-估计的厚尾序列结构变点检验研究[D]. 西安科技大学, 2021.

[29] Phillips P. C. B.. Econometric Theory [J]. Econometric Theory , 1990, 6: 44-62.

[30] Knight,K..Limit Theory for M-Estimates in an Integrated Infinite Variance[J].Econometric Theory, 1991, 7(2): 200-212.

[31] Phillips, Peter C.B. and Victor Solo. Asymptotics for Linear Processes[J]. Annals of Statistics, 1992, 20: 971-1001.

[32] Horváth, L, Kokoszka, P.. A Bootstrap Approximation to a Unit Root Test Statistic for Heavy-tailed Observations[J].Statistics & Probability Letters, 2003, 63(2): 163-173.

[33] Nolan J P. Numerical calculation of stable densities and distribution functions[J]. Communications in Statistics Stochastic Models, 1997, 13(4): 759-7.

[34] Cindy Shin-Huei Wang and Yi Meng Xie. Structural Change and Monitoring Tests[M]. Handbook of Financial Econometrics and Statistics, 2015.

[35] Steland, A.. Monitoring procedures to detect unit roots and stationarity[J]. Economic Theory, 2007, 23(6): 1108-1135.

[36] Zhanshou Chen, Zheng Tian, Yuesong Wei. Monitoring change in persistence in linear time series[J]. Statistics & Probability Letters, 2010, 79(19-20): 1520-1527.

[37] 秦瑞兵, 孙丽, 宋冠仪. 线性回归模型系数变点的在线监测[J]. 陕西科技大学学报, 2020, 38(1): 175-179.

[38] Qi P Y, Jin Z, Tian Z, et al. Monitoring persistent change in a heavy-tailed sequence with polynomial trends[J]. Journal of the Korean Statistical Society, 2013, 42(4): 497-506.

[39] Dominik Wied, Pedro Galeano. Monitoring correlation change in a sequence of random variables[J]. Journal of Statistical Planning and Inference, 2013, 143(1): 186-196.

[40] Hušková M. Tests and Estimators for Change Point Problem Based on M-Statistics[J]. Stat. Decis, 1996, 14: 115-136.

[41] Hušková M. M-Procedures for Detection of Changes for Dependent Observations[J]. Communications in Statistics, B.Simulation and Computation, 2012, 41: 1032-1050.

[42] Qin R, Liu W. A Robust Test for Mean Change in Dependent Observations[J]. Journal of Inequalities and Applications, 2015, 48: 293-304.

[43] Qin R, Liu W. Ratio Detection for Mean Change in α Mixing Observations[J]. Communications in Statistics, 2019, 48(7): 1693-1708.

[44] Sohrabi M, Zarepour M. Asymptotic Theory for M-Estimates in Unstable AR(p) Processes with Infinite Variance Innovations[J]. Journal of Time, 2015, 34(3): 362-367.

[45] BOUDT, K., CROUX, C.. Robust M-estimation of multivariate GARCH models[J]. Computational statistics & data analysis, 2010, 54(11): 2459-2469.

[46] Lei L, Bickel P J, Karoui N E.Asymptotics for High Dimensional Regression M-Estimates:Fixed Design Results[J]. Probability Theory and Related Fields, 2016(3): 1-97.

[47] Sen A, Srivastava, et al. On ests for Detecting Change in Mean[J]. Annals of the Institute of Statistical Mathematics, 1975, 27(1): 479-486.

[48] Douglas M. Hawkins. Testing a Sequence of Observations for a Shift in Location[J]. Journal of the American Statistical Association, 1977, 72(357): 180-186.

[49] Yao,Yi-Ching. Approximating the Distribution of the Maximum Likelihood Estimate of the Change-Point in a Sequence of Independent Random Variables[J]. The Annals of Statistics, 1987, 3(3): 1321-1328.

[50] Bai J. Least Squares Estimation of a Shift in Linear Process[J]. Journal of Time Series Analysis, 2010, 15(5): 453-472.

[51] Kim S, Cho S, Lee S. On the Cusum Test for Parameter Changes in GARCH(1,1)Models [J]. Communication in Statistics, 2000, 29(3): 445-462.

[52] Ibbotson R G, Jaffe J F. The Behavior of Stock Market Prices[J]. Journal of Finance, 2006, 30(4): 1027-1042.

[53] 张尧庭. 厚尾分布及其在股市中的应用[J]. 中国货币市场, 2002, (3): 42-43.

[54] Weitzman M L. Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change Some Empirical Examples of Deep Structural Un-certainty about Climate Extremes[J]. 2011, 28(2): 18-24.

[55] 刘昌义, 潘家华. 气候变化的不确定性及其经济影响与政策含义[J]. 中国人口资源与环境, 2012, 22(11): 15-20.

[56] Barabasi A L. The Origin of Bursts and Heavy Tails in Human Dynamics[J]. Nature, 2005, 435(7039): 207-211.

[57] 卓志, 王伟哲. 巨灾风险厚尾: POT模型及其应用[J]. 保险研究, 2011, (8): 13-19.

[58] Brodin E, Kluppelberg C. Extreme Value Theory in Finance[M]. Everitt, B. and Meiinick, E. Encyclopedia of Quantitive Risk Assessment, 2007.

[59] Allan R, Soden B.Atmospheric Warming and the Amplification of Precipitation Extremes[J]. Science, 2008, 321(5895): 1481-1484.

[60] 金浩, 高奎, 张思. 基于Bootstrap方法的重尾相依序列均值变点Ratio检验[J]. 统计与决策, 2019, 35(23): 11-16.

[61] Peštová B, Pešta M. Abrupt Change in Mean Using Block Bootstrap and Avoiding Variance Estimation[J]. Computational Stats, 2018, 33(1): 413-441.

[62] Perron P. Dealing with Structural Breaks[M]. Working Paper Series, Department of economics, Boston University, 2005.

[63] Csorgo M, Horvath L. Limit Theorem in Change-Point Analysis[M]. New York: Wiley Series in Probability and Statistics, John Wiley & Sons, 1997.

[64] Chen J, Gupta A K. Parametric Statistical Change Point Analysis[M]. Boston: Birkha-user, 2000.

[65] Jin H, Zhang S, Zhang J, et al. Bootstrap Procedures for Variance Breaks Test in Time Series with a Changing Trend[J]. Communications in Statistics, 2018, 47(16): 4609-4627.

[66] Dong C, Tan C, Jin B, et al. Inference on the Change Point Estimator of Variance in Measurement Error Models[J]. Lithuanian Mathematical Journal, 2016, 56(4): 1-18.

[67] Chen Z, Xing Y, Li F. Sieve Bootstrap Monitoring for Change from Short to Long Memory[J]. Economics Letters, 2016, 140: 53-56.

[68] Belaire-Franch J, Kim J Y, Amador R B. Corrigendum to “Detection of Change in Persistence of a Linear Time Series”[J]. Journal of Econometrics, 2002, 109(2): 389-392.

[69] 陈希孺. 变点统计分析简介[J]. 数理统计与管理, 1991, 10(2): 52-59.

[70] Bai J. Least Squares Estimation of a Shift in Linear Processes[J]. Journal of Time, 2010, 15(5): 453-472.

[71] Lavielle M, Moulines E. Least Squares Estimation of an Unknown Number of Shifts in Time Series[J]. Journal of Time Series Analysis, 2000, 21(1): 33-59.

[72] Kokoszka P, Leipus R. Change-Point in the Mean of Dependent Observations[J]. Statistics and Probability Letters, 1998, 40(4): 385-393.

[73] Taylor R, Leybourne, Stephen J. Persistence Change Tests and Shifting Stable Auto-Regressions[J]. Economics Letter, 2006, 91(1): 44-49.

[74] Horvath L, Horvath Z, Hukova M. Ratio Tests for Change Point Detection[J]. Institute of Mathematical Statistics, 2008, 1: 293-304.

[75] 赵文芝, 吕会琴. 厚尾相依序列均值变点Ratio检验[J]. 山西大学学报, 2016, 39(3): 410-414.

[76] De Jong, R.M. and Davidson J. The Functional Central Limit Theorem and Weak Convergence to Stochastic Integrals I: Weakly Dependent and Fractionally Integrated Processes [J]. Econometric Theory, 2000, 16: 621-642.

[77] RMD Jong, Amsler C, Schmidt P. A Robust Version of the KPSS Test Based on Indicators[J]. Journal of Econometrics, 2007, 137(2): 311-333.

[78] Fan J, Yao Q. Nonlinear Time Series: Nonparametric and Parametric Methods[M]. 科学出版社, 2006.

[79] Davis Richard A., Knight Keith, Liu Jian. M-estimation for autoregressions with infinite variance[J]. Stochastic Processes and their Applications, 1992, 40(1): 145-180.

[80] Keith Knight. Limit Theory for M-Estimates in an Integrated Infinite Variance Process[J]. Econometric Theory, 1991, 7(2): 133-142.

[81] Maryam Sohrabi, Mahmoud Zarepour. Asymptotic theory for M-estimates in unstable AR(p) processes with infinite variance innovations[J]. Journal of Statistical Planning and Inference, 2019, 198: 105-11.