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CN 34-1054/N

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Convergence rate of location parameter change-point estimator

  • 1.

    School of Mathematics, Hefei University of Technology, Hefei 230009, China

  • 2.

    School of Mathematical Sciences, Xinjiang Normal University, Urumqi 830054, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.03.006
  • Received Date: 06 January 2014
  • Accepted Date: 25 April 2014
  • Rev Recd Date: 25 April 2014
  • Publish Date: 30 March 2015
    Abstract

  • Abstract

    For the location parameter change point model allowing at most one change, an antisymmetric kernel function was constructed and a change point estimator was proposed with the help of the CUSUM method. The consistency and convergence rate of the change point estimator were studied under normal conditions and its consistency was given under the condition of local alternative hypothesis. Finally, simulation was performed under different sample sizes and parameter settings. From the histograms, the closer the change-point location gets to the middle, the more accurate the estimate is. The results show that our methods are effective.

    Abstract

    For the location parameter change point model allowing at most one change, an antisymmetric kernel function was constructed and a change point estimator was proposed with the help of the CUSUM method. The consistency and convergence rate of the change point estimator were studied under normal conditions and its consistency was given under the condition of local alternative hypothesis. Finally, simulation was performed under different sample sizes and parameter settings. From the histograms, the closer the change-point location gets to the middle, the more accurate the estimate is. The results show that our methods are effective.
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    • References(15)
  • [1]
    Bhattacharya P K, Frierson F J. A nonparametric control chart for detecting small disorders [J]. The Annals of Statistics, 1981:9,544-554.
    [2]
    Miao Baiqi. Nonparametric methods for a model with at most one change point[J]. J Sys Sci & Math Scis, 1993,13(2): 132-140.
    缪柏其.关于只有一个变点模型的非参数推断[J]. 系统科学与数学,1993,13(2): 132-140.
    [3]
    Gombay E,Horvth L. An application of U-statistics to change-point analysis[J]. Acta Sci Math (Szeged), 1995, 60: 345-357.
    [4]
    Gombay E. U-statistics for change under alternatives[J]. Journal of Multivariate Analysis, 2001, 78:139-158.
    [5]
    Csrg M, Horvth L. Limit Theorems in Change-Points Analysis[M]. New York: John Wily and Sons,1997.
    [6]
    Bassevile M, Nikiforov I V. Detection of Abrupt Changes: Theory and Application[M]. Englewood Cliffs, NJ: Prentice Hall, 1993.
    [7]
    Brodsky B E, Darkhovsky B S. Nonparametric Methods in Change-Point Problems[M]. Netherlands: Kluwer Academic Publishers, 1993.
    [8]
    Chen J, Gupta A K. Parametric Statistical Change Point Analysis[M].New York: Springer, 2000.
    [9]
    Csrg M, Horvth L. Nonparametric methods for change point problems[C]// Handbook of Statistics, Control and Reliability. New York: North-Holland, 1988.
    [10]
    Krishnaiah P R, Miao B Q. Review about estimates of change-point[J]. Handbook of Statistics, 1988,7:375-402.
    [11]
    Miao Baiqi, Wei Denyun. Nonparameter inferences on the change point of a scale parameter[J]. Journal of University of Science and Technology of China, 1994, 24(3): 263-270.
    缪柏其,魏登云.关于刻度参数变点的非参数统计推断[J].中国科学技术大学学报,1994,24(3): 263-270.
    [12]
    Horvth L, Hukov M. Testing for changes using permutations of U-statistics[J]. Journal of Statistical Planning and Inference, 2005, 128: 351-371.
    [13]
    Miao B Q, Zhang S G. The exponential convergence rates of the projection residue of U-statistics and their application[J]. Chinese Journal of Applied Probability, 1994,10(3):253-264.
    [14]
    Bennett G. Probability inequalities for the sums of independent random variables[J]. Journal of American Statistics Association, 1962, 57: 33-57.
    [15]
    林正炎，白志东. 概率不等式[M]. 北京：科学出版社, 2006.
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    [1]
    Bhattacharya P K, Frierson F J. A nonparametric control chart for detecting small disorders [J]. The Annals of Statistics, 1981:9,544-554.
    [2]
    Miao Baiqi. Nonparametric methods for a model with at most one change point[J]. J Sys Sci & Math Scis, 1993,13(2): 132-140.
    缪柏其.关于只有一个变点模型的非参数推断[J]. 系统科学与数学,1993,13(2): 132-140.
    [3]
    Gombay E,Horvth L. An application of U-statistics to change-point analysis[J]. Acta Sci Math (Szeged), 1995, 60: 345-357.
    [4]
    Gombay E. U-statistics for change under alternatives[J]. Journal of Multivariate Analysis, 2001, 78:139-158.
    [5]
    Csrg M, Horvth L. Limit Theorems in Change-Points Analysis[M]. New York: John Wily and Sons,1997.
    [6]
    Bassevile M, Nikiforov I V. Detection of Abrupt Changes: Theory and Application[M]. Englewood Cliffs, NJ: Prentice Hall, 1993.
    [7]
    Brodsky B E, Darkhovsky B S. Nonparametric Methods in Change-Point Problems[M]. Netherlands: Kluwer Academic Publishers, 1993.
    [8]
    Chen J, Gupta A K. Parametric Statistical Change Point Analysis[M].New York: Springer, 2000.
    [9]
    Csrg M, Horvth L. Nonparametric methods for change point problems[C]// Handbook of Statistics, Control and Reliability. New York: North-Holland, 1988.
    [10]
    Krishnaiah P R, Miao B Q. Review about estimates of change-point[J]. Handbook of Statistics, 1988,7:375-402.
    [11]
    Miao Baiqi, Wei Denyun. Nonparameter inferences on the change point of a scale parameter[J]. Journal of University of Science and Technology of China, 1994, 24(3): 263-270.
    缪柏其,魏登云.关于刻度参数变点的非参数统计推断[J].中国科学技术大学学报,1994,24(3): 263-270.
    [12]
    Horvth L, Hukov M. Testing for changes using permutations of U-statistics[J]. Journal of Statistical Planning and Inference, 2005, 128: 351-371.
    [13]
    Miao B Q, Zhang S G. The exponential convergence rates of the projection residue of U-statistics and their application[J]. Chinese Journal of Applied Probability, 1994,10(3):253-264.
    [14]
    Bennett G. Probability inequalities for the sums of independent random variables[J]. Journal of American Statistics Association, 1962, 57: 33-57.
    [15]
    林正炎，白志东. 概率不等式[M]. 北京：科学出版社, 2006.
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