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Strong convergence rate of estimators of change point and its application

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  • Shi, Xiaoping
  • Wu, Yuehua
  • Miao, Baiqi

Abstract

Let {Xn,n[greater-or-equal, slanted]1} be an independent sequence with a mean shift. We consider the cumulative sum (CUSUM) estimator of a change point. It is shown that, when the rth moment of Xn is finite, for n[greater-or-equal, slanted]1 and r>1, strong convergence rate of the change point estimator is o(M(n)/n), for any M(n) satisfying that M(n)[short up arrow][infinity], which has improved the results in the literature. Furthermore, it is also shown that the preceding rate is still valid for some dependent or negative associate cases. We also propose an iterative algorithm to search for the location of a change point. A simulation study on a mean shift model with a stable distribution is provided, which demonstrates that the algorithm is efficient. In addition, a real data example is given for illustration.

Suggested Citation

  • Shi, Xiaoping & Wu, Yuehua & Miao, Baiqi, 2009. "Strong convergence rate of estimators of change point and its application," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 990-998, February.
    • Handle: RePEc:eee:csdana:v:53:y:2009:i:4:p:990-998
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    References listed on IDEAS

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    1. Jushan Bai, 1994. "Least Squares Estimation Of A Shift In Linear Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 15(5), pages 453-472, September.
    2. Kokoszka, Piotr & Leipus, Remigijus, 1998. "Change-point in the mean of dependent observations," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 385-393, November.
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    4. Lavielle, Marc, 1999. "Detection of multiple changes in a sequence of dependent variables," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 79-102, September.
    5. Matula, Przemyslaw, 1992. "A note on the almost sure convergence of sums of negatively dependent random variables," Statistics & Probability Letters, Elsevier, vol. 15(3), pages 209-213, October.
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    Cited by:

    1. Li Zhaoyuan & Tian Maozai, 2017. "Detecting Change-Point via Saddlepoint Approximations," Journal of Systems Science and Information, De Gruyter, vol. 5(1), pages 48-73, February.
    2. Saisai Ding & Xiaoqin Li & Xiang Dong & Wenzhi Yang, 2020. "The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application," Mathematics, MDPI, vol. 8(12), pages 1-12, November.
    3. Shi, Xiaoping & Wu, Yuehua & Miao, Baiqi, 2009. "A note on the convergence rate of the kernel density estimator of the mode," Statistics & Probability Letters, Elsevier, vol. 79(17), pages 1866-1871, September.
    4. Qin, Ruibing & Tian, Zheng & Jin, Hao & Zhang, Xiaowei, 2010. "Strong convergence rate of robust estimator of change point," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(10), pages 2026-2032.
    5. Eunju Hwang & Dong Shin, 2016. "Kernel estimators of mode under $$\psi $$ ψ -weak dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 301-327, April.

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