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On nonparametric change point estimator based on empirical characteristic functions

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Abstract

We propose a nonparametric change point estimator in the distributions of a sequence of independent observations in terms of the test statistics given by Hušková and Meintanis (2006) that are based on weighted empirical characteristic functions. The weight function ω(t; a) under consideration includes the two weight functions from Hušková and Meintanis (2006) plus the weight function used by Matteson and James (2014), where a is a tuning parameter. Under the local alternative hypothesis, we establish the consistency, convergence rate, and asymptotic distribution of this change point estimator which is the maxima of a two-side Brownian motion with a drift. Since the performance of the change point estimator depends on a in use, we thus propose an algorithm for choosing an appropriate value of a, denoted by a s which is also justified. Our simulation study shows that the change point estimate obtained by using a s has a satisfactory performance. We also apply our method to a real dataset.

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References

  1. Bai J. Estimation of a change point in multiple regression models. Rev Econ Stat, 1997, 79: 551–63

    Article  Google Scholar 

  2. Balke N S. Detecting level shifts in time series. J Bus Econom Statist, 1993, 11: 81–92

    Google Scholar 

  3. Brodskij B E, Darchovskij B S. Non-parametric Statistical Diagnosis: Problems and Methods. Dordrecht: Kluwer Academic Publishers, 2000

    Book  Google Scholar 

  4. Carlstein E. Nonparametric change point estimation. Ann Stat, 1988, 16: 188–197

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen J, Gupta A K. Parametric Statistical Change Point Analysis: With Applications to Genetics, Medicine, and Finance. New York: Springer Science & Business Media, 2011

    MATH  Google Scholar 

  6. Chen X R. Testing and interval estimation in a change-point model allowing at most one change. Sci China Ser A, 1988, 30: 817–827

    Google Scholar 

  7. Cobb G W. The problem of the Nile: Conditional solution to a change point problem. Biometrika, 1978, 65: 243–251

    Article  MathSciNet  MATH  Google Scholar 

  8. Csörgo M, Horváth L. Limit Theorems in Change-Point Analysis. New York: John Wiley & Sons Inc, 1997

    MATH  Google Scholar 

  9. Dumbgen L. The asymptotic behavior of some nonparametric change point estimators. Ann Stat, 1991, 19: 1471–1495

    Article  MathSciNet  MATH  Google Scholar 

  10. Epps T W. Limiting behaviour of the ICF test for normality under Gram-Charlier alternatives. Statist Probab Lett, 1999, 42: 175–184

    Article  MathSciNet  MATH  Google Scholar 

  11. Feuerverger A, Mureika R A. The empirical characteristic function and its applications. Ann Stat, 1977, 5: 88–97

    Article  MathSciNet  MATH  Google Scholar 

  12. Gombay E. U-statistics for change under alternatives. J Multivariate Anal, 2001, 78: 139–158

    Article  MathSciNet  MATH  Google Scholar 

  13. Hinkley D. Inference about the change-point in a sequence of random variables. Biometrika, 1970, 57: 1–17

    Article  MathSciNet  MATH  Google Scholar 

  14. Holmes M, Kojadinovic I, Quessy J. Nonparametric tests for change-point detection á la Gombay and Horváth. J Multivar Anal, 2013, 115: 16–32

    Article  MathSciNet  MATH  Google Scholar 

  15. Husková M, Meintanis S G. Change point analysis based on empirical characteristic functions. Metrika, 2006, 6: 145–168

    Article  MathSciNet  MATH  Google Scholar 

  16. Jin B S, Dong C L, Tan C C, et al. Estimator of a change point in single index models. Sci China Math, 2014, 57: 1701–1712

    Article  MathSciNet  MATH  Google Scholar 

  17. Kankainen A, Ushakov N G. A consistent modification of a test for independence based on the empirical characteristic function. J Math Sci, 1998, 89: 1486–1494

    Article  MathSciNet  MATH  Google Scholar 

  18. Kent J T. A weak convergence theorem for the empirical characteristic function. J Appl Probab, 1975, 12: 515–523

    Article  MathSciNet  MATH  Google Scholar 

  19. Koutrouvelis I A, Meintanis S G. Testing for stability based on the empirical characteristic function with applications to financial data. J Stat Comput Simul, 1999, 64: 275–300

    Article  MATH  Google Scholar 

  20. Matteson D S, James N A. A nonparametric approach for multiple change point analysis of multivariate data. J Amer Statist Assoc, 2014, 109: 334–345

    Article  MathSciNet  Google Scholar 

  21. Parzen E. On estimation of a probability density function and mode. Ann Math Stat, 1962, 33: 1065–1076

    Article  MathSciNet  MATH  Google Scholar 

  22. Rafajlowicz E, Pawlak M, Steland A. Nonparametric sequential change-point detection by a vertically trimmed box method. IEEE Trans Inform Theory, 2010, 56: 3621–3634

    Article  MathSciNet  Google Scholar 

  23. Shi X P, Wu Y H, Miao B Q. Strong convergence rate of estimators of change point and its application. Comput Stat Data Anal, 2009, 53: 990–998

    Article  MathSciNet  MATH  Google Scholar 

  24. Ushakov N G. Selected Topics in Characteristic Functions. Berlin: Walter de Gruyter, 1999

    Book  MATH  Google Scholar 

  25. Zeileis A, Kleiber C, Krämer W, et al. Testing and dating of structural changes in practice. Comput Statist Data Anal, 2003, 44: 109–123

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    ChangChun Tan

  2. Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, BC V2C 0C8, Canada

    XiaoPing Shi

  3. Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada

    XiaoYing Sun & YueHua Wu

Authors
  1. ChangChun Tan
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  2. XiaoPing Shi
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  3. XiaoYing Sun
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  4. YueHua Wu
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Correspondence to YueHua Wu.

Additional information

In memory of Professor Xiru Chen (1934–2005)

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Tan, C., Shi, X., Sun, X. et al. On nonparametric change point estimator based on empirical characteristic functions. Sci. China Math. 59, 2463–2484 (2016). https://doi.org/10.1007/s11425-016-0138-x

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