Communications for Statistical Applications and Methods

  • 제24권5호
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  • Pages.481-492
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  • 2017
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  • 2287-7843(pISSN)
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  • 2383-4757(eISSN)
한국통계학회 (The Korean Statistical Society)
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Convergence rate of a test statistics observed by the longitudinal data with long memory

  • Kim, Yoon Tae (Department of Finance and Information Statistics, Hallym University) ;
  • Park, Hyun Suk (Department of Finance and Information Statistics, Hallym University)
  • 투고 : 2017.05.25
  • 심사 : 2017.08.29
  • 발행 : 2017.09.30
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초록

This paper investigates a convergence rate of a test statistics given by two scale sampling method based on $A\ddot{i}t$-Sahalia and Jacod (Annals of Statistics, 37, 184-222, 2009). This statistics tests for longitudinal data having the existence of long memory dependence driven by fractional Brownian motion with Hurst parameter $H{\in}(1/2,\;1)$. We obtain an upper bound in the Kolmogorov distance for normal approximation of this test statistic. As a main tool for our works, the recent results in Nourdin and Peccati (Probability Theory and Related Fields, 145, 75-118, 2009; Annals of Probability, 37, 2231-2261, 2009) will be used. These results are obtained by employing techniques based on the combination between Malliavin calculus and Stein's method for normal approximation.

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과제정보

연구 과제 주관 기관 : National Research Foundation of Korea

참고문헌

  1. Ait-Sahalia Y and Jacod J (2009). Testing for jumps in a discretely observed process. The Annals of Statistics, 37, 184-222. https://doi.org/10.1214/07-AOS568
  2. Beran J (1994). Statistics for Long-Memory Processes, Chapman and Hall, London.
  3. Kim YT and Park HS (2015). Estimation of Hurst parameter in the longitudinal data with long memory. Communications for Statistical Applications and Methods, 22, 295-304. https://doi.org/10.5351/CSAM.2015.22.3.295
  4. Kim YT and Park HS (2016). Berry-Esseen Type bound of a sequence {$\frac{X_N}{Y_N}$} and its application. Journal of the Korean Statistical Society, 45, 544-556. https://doi.org/10.1016/j.jkss.2016.03.004
  5. Kim YT and Park HS (2017a). Optimal Berry-Esseen bound for statistical estimations and its appli-cation to SPDE. Journal of Multivariate Analysis, 155, 284-304. https://doi.org/10.1016/j.jmva.2017.01.006
  6. Kim YT and Park HS (2017b). Optimal Berry-Esseen bound for an estimator of parameter in the Ornstein-Uhlenbeck process. Journal of the Korean Statistical Society, 46, 413-425. https://doi.org/10.1016/j.jkss.2017.01.002
  7. Michel R and Pfanzagl J (1971). The accuracy of the normal approximation for minimum contrast estimates. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 18, 73-84. https://doi.org/10.1007/BF00538488
  8. Nourdin I (2013). Lectures on Gaussian approximations with Malliavin calculus. Seminaire de Probabilit es, 45, 3-89.
  9. Nourdin I and Peccati G (2009a). Stein’s method on Wiener chaos. Probability Theory and Related Fields, 145, 75-118. https://doi.org/10.1007/s00440-008-0162-x
  10. Nourdin I and Peccati G (2009b). Stein's method and exact Berry-Esseen asymptotics for functionals for functionals of Gaussian fields, The Annals of Probability, 37, 2231-2261. https://doi.org/10.1214/09-AOP461
  11. Nualart D (2006). Malliavin Calculus and Related Topics (2nd ed), Springer, Berlin.