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The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application

Author

Listed:
  • Saisai Ding

    (School of Mathematical Sciences, Anhui University, Hefei 230601, China)

  • Xiaoqin Li

    (School of Mathematical Sciences, Anhui University, Hefei 230601, China)

  • Xiang Dong

    (School of Life Sciences, Anhui University, Hefei 230601, China)

  • Wenzhi Yang

    (School of Mathematical Sciences, Anhui University, Hefei 230601, China)

Abstract

In this paper, we investigate the CUSUM-type estimator of mean change-point models based on m -asymptotically almost negatively associated ( m -AANA) sequences. The family of m -AANA sequences contains AANA, NA, m -NA, and independent sequences as special cases. Under some weak conditions, some convergence rates are obtained such as O P ( n 1 / p − 1 ) , O P ( n 1 / p − 1 log 1 / p n ) and O P ( n α − 1 ) , where 0 ≤ α < 1 and 1 < p ≤ 2 . Our rates are better than the ones obtained by Kokoszka and Leipus (Stat. Probab. Lett., 1998, 40, 385–393). In order to illustrate our results, we do perform simulations based on m -AANA sequences. As important applications, we use the CUSUM-type estimator to do the change-point analysis based on three real data such as Quebec temperature, Nile flow, and stock returns for Tesla. Some potential applications to change-point models in finance and economics are also discussed in this paper.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2113-:d:451374
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    References listed on IDEAS

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    1. Lajos Horváth & Gregory Rice, 2014. "Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 219-255, June.
    2. Zeileis, Achim & Kleiber, Christian & Kramer, Walter & Hornik, Kurt, 2003. "Testing and dating of structural changes in practice," Computational Statistics & Data Analysis, Elsevier, vol. 44(1-2), pages 109-123, October.
    3. Michael Messer & Stefan Albert & Gaby Schneider, 2018. "The multiple filter test for change point detection in time series," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(6), pages 589-607, August.
    4. Mengmei Xi & Xin Deng & Xuejun Wang & Zhaoyang Cheng, 2018. "Lp convergence and complete convergence for weighted sums of AANA random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(22), pages 5604-5613, November.
    5. 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.
    6. Sangyeol Lee & Jeongcheol Ha & Okyoung Na & Seongryong Na, 2003. "The Cusum Test for Parameter Change in Time Series Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(4), pages 781-796, December.
    7. Lajos Horváth & Gregory Rice, 2014. "Rejoinder on: Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 287-290, June.
    8. Kokoszka, Piotr & Leipus, Remigijus, 1998. "Change-point in the mean of dependent observations," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 385-393, November.
    9. Trinh Hoai Nam & Tien-Chung Hu & Andrei Volodin, 2017. "Maximal inequalities and strong law of large numbers for sequences of m-asymptotically almost negatively associated random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(6), pages 2696-2707, March.
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