Advertisement

Statistical Papers

, 51:57 | Cite as

Kernel type smoothed quantile estimation under long memory

  • Lihong WangEmail author
Regular Article

Abstract

This paper studies nonparametric kernel type (smoothed) estimation of quantiles for long memory stationary sequences. The uniform strong consistency and asymptotic normality of the estimates with rates are established. Finite sample behaviors are investigated in a small Monte Carlo simulation study.

Keywords

Asymptotic normality Long memory time series Quantile estimation Strong consistency 

Mathematics Subject Classification (2000)

62M10 62G05 62G20 

References

  1. 1.
    Avram F and Taqqu MS (1987). Noncentral limit theorems and Appell polynomials. Ann Probab 15: 767–775 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beran J (1994). Statistics for long-memory processes. Chapman and Hall, New York zbMATHGoogle Scholar
  3. 3.
    Cai ZW and Roussas GG (1992). Uniform strong estimation under β-mixing with rates. Stat Probab Lett 15: 47–55 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cai ZW and Roussas GG (1997). Smooth estimate of quantiles under association. Stat Probab Lett 36: 275–287 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Csörgő S and Mielniczuk J (1995a). Density estimation under long-range dependence. Ann Stat 23: 990–999 CrossRefGoogle Scholar
  6. 6.
    Csörgő S and Mielniczuk J (1995b). Nonparametric regression under long-range dependent normal errors. Ann Stat 23: 1000–1014 CrossRefGoogle Scholar
  7. 7.
    Csörgő M, Szyszkowicz B and Wang L (2006). Strong invariance principles for sequential Bahadur–Kiefer and Vervaat error processes of long-range dependent sequences. Ann Stat 34: 1013–1044 CrossRefGoogle Scholar
  8. 8.
    Davydov YA (1970). The invariance principle for stationary processes. Theory Probab Appl 15: 487–498 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Dehling H and Taqqu MS (1989). The empirical process of some long-range dependent sequences with an application to U-statistics. Ann Stat 17: 1767–1783 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Doukhan P, Oppenheim G and Taqqu MS (2003). Theory and applications of long-range dependence. Birkhäuser, Boston zbMATHGoogle Scholar
  11. 11.
    Draghicescu D, Ghosh S (2003) Smooth nonparametric quantiles. In: Proceedings, MENP-2, 22–27 April 2002. Geometry Balkan Press, Bucharest, pp 45–52Google Scholar
  12. 12.
    Estévez G and Vieu P (2003). Nonparametric estimation under long-memory dependence. J Nonparametric Stat 15: 535–551 zbMATHCrossRefGoogle Scholar
  13. 13.
    Falk M (1984). Relative deficiency of kernel type estimators of quantiles. Ann Stat 12: 261–268 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Falk M (1985). Asymptotic normality of the kernel quantile estimator. Ann Stat 13: 428–433 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Falk M and Reiss RD (1989). Weak convergence of smoothed and nonsmoothed bootstrap quantile estimates. Ann Probab 17: 362–371 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ghosh S, Beran J and Innes J (1997). Nonparametric conditional quantile estimation in the presence of long memory. Student 2: 109–117 Google Scholar
  17. 17.
    Ghosh S, Draghicescu D (2002) An algorithm for optimal bandwidth selection for smooth nonparametric quantile estimation. Statistical data analysis based on the L 1 norm and related methods, pp 161–168Google Scholar
  18. 18.
    Granger CW and Joyeux R (1980). An introduction to long-range time series models and fractional differencing. J Time Ser Anal 1: 15–30 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Guégan D (2005). How can we define the concept of long memory? An econometric survey. Economet Rev 24: 113–149 zbMATHCrossRefGoogle Scholar
  20. 20.
    Ho HC and Hsing T (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. Ann Stat 24: 992–1024 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hosking JRM (1981). Fractional differencing. Biometrika 68: 165–176 zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lejeune M and Sarda P (1991). Smooth estimators of distribution density functions. Comp Stat Data Anal 14: 457–471 CrossRefMathSciNetGoogle Scholar
  23. 23.
    Lobato I and Robinson PM (1996). Averaged periodogram estimation of long-memory. J Economet 73: 303–324 zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Moulines E and Soulier P (2003). Semiparametric spectral estimation for fractional processes. In: Doukhan, P, Oppenheim, G and Taqqu, MS (eds) Theory and applications of long-range dependence, pp 251–301. Birkhäuser, Boston Google Scholar
  25. 25.
    Nadaraya EA (1964). Some new esimates for distribution function. Theory Probab Appl 9: 497–500 CrossRefGoogle Scholar
  26. 26.
    Parzen E (1979). Nonparametric statistical data modeling. J Am Stat Assoc 74: 105–121 zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ralescu SS and Sun S (1993). Necessary and sufficient conditions for the asymptotic normality perturbed sample quantiles. J Stat Plan Inference 35: 55–64 zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Read RR (1972). The asymptotic inadmissibility of the sample distribution function. Ann Math Stat 43: 89–95 zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Reiss RD (1980). Estimation of quantiles in certain nonparametric models. Ann Stat 8: 87–105 zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Robinson PM (1994). Semiparametric analysis of long-memory time series. Ann Stat 22: 515–539 zbMATHCrossRefGoogle Scholar
  31. 31.
    Robinson PM (1995). Log-periodogram regression of time series with long range dependence. Ann Stat 23: 1048–1074 zbMATHCrossRefGoogle Scholar
  32. 32.
    Robinson PM (1995). Gaussian semiparametric estimation of long range dependence. Ann Stat 23: 1630–1661 zbMATHCrossRefGoogle Scholar
  33. 33.
    Wang Q, Lin Y and Gulati M (2003). Strong approximation for long-memory processes with applications. J Theor Probab 16: 377–389 zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Yamato H (1973). Uniform convergence of an estimator of a distribution function. Bull Math Stat 15: 69–78 zbMATHMathSciNetGoogle Scholar
  35. 35.
    Youndjé É and Vieu P (2006). A note on quantile estimation for long-range dependent stochastic processes. Stat Probab Lett 76: 109–116 zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematics, Institute of Mathematical ScienceNanjing UniversityNanjingPeople’s Republic of China

Personalised recommendations