Skip to main content
SpringerLink
Log in

Edgeworth expansion for the kernel quantile estimator

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Using the kernel estimator of the pth quantile of a distribution brings about an improvement in comparison to the sample quantile estimator. The size and order of this improvement is revealed when studying the Edgeworth expansion of the kernel estimator. Using one more term beyond the normal approximation significantly improves the accuracy for small to moderate samples. The investigation is non- standard since the influence function of the resulting L-statistic explicitly depends on the sample size. We obtain the expansion, justify its validity and demonstrate the numerical gains in using it.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bickel P., Götze F., van Zwet W. (1986) The Edgeworth expansion for U-statistics of degree two. The Annals of Statistics 14: 1463–1484

    Article  MATH  MathSciNet  Google Scholar 

  • Borovkov K., Dehling H., Pfeifer D. (1996) On asymptotic behavior of weighted sample quantiles. Mathematical Methods of Statistics 5: 173–186

    MATH  MathSciNet  Google Scholar 

  • Callaert H., Janssen P., Veraverbeke N. (1980) An Edgeworth expansion for U-statistics. The Annals of Statistics 8: 299–312

    Article  MATH  MathSciNet  Google Scholar 

  • Dharmadhikari S., Fabian V., Jogdeo K. (1968) Bounds on the moments of martingales. The Annals of Mathematical Statistics 39: 1719–1723

    Article  MATH  MathSciNet  Google Scholar 

  • DiDonato A., Morris A. (1987) ALGORITHM 654: FORTRAN subroutines for computing the incomplete gamma function ratios and their inverse. Association for Computing Machinery Transactions on Mathematical Software 13(3): 318–319

    Article  MATH  Google Scholar 

  • Falk M. (1984) Relative deficiency of kernel type estimators of quantiles. The Annals of Statistics 12(1): 261–268

    Article  MATH  MathSciNet  Google Scholar 

  • Falk M. (1985) Asymptotic normality of the kernel quantile estimator. The Annals of Statistics 13(1): 428–433

    Article  MATH  MathSciNet  Google Scholar 

  • Feller W. (1971) An introduction to probability and its applications (Vol. 2). Wiley, New York

    Google Scholar 

  • Helmers R. (1982) Edgeworth expansions for linear combinations of order statistics. Mathematical Centre Tracts (Vol. 105). Amsterdam, Mathematisch Centrum

    Google Scholar 

  • Jing B.-Y., Wang Q. (2003) Edgeworth expansions for U-statistics under minimal conditions. The Annals of Statistics 31(4): 1376–1391

    Article  MATH  MathSciNet  Google Scholar 

  • Kozek A. (2005) How to vombine M-estimators to estimate quantiles and a score function. Sankhya 67(2): 277–294

    MATH  MathSciNet  Google Scholar 

  • Lai T., Wang J. (1993) Edgeworth expansions for symmetric statistics with applications to bootstrap methods. Statistica Sinica 3: 517–542

    MATH  MathSciNet  Google Scholar 

  • Lorenz M. (1905) Methods of measuring concentration of wealth. Journal of the American Statistical Association 9: 209–219

    Article  Google Scholar 

  • Maesono Y. (1987) Edgeworth expansion for one-sample U-statistics. Bulletin of Informatics and Cybernetics 22: 189–197

    MATH  MathSciNet  Google Scholar 

  • Ogryczak W., Ruszczyński A. (2002) Dual stochastic dominance and related mean-risk models. SIAM Journal of Optimization 13(1): 60–78

    Article  MATH  Google Scholar 

  • Serfling R. (1980) Approximation theorems of mathematical statistics. Wiley, New York

    Book  MATH  Google Scholar 

  • Sheather S., Marron J. (1990) Kernel quantile estimators. Journal of the American Statistical Association 85(410): 410–416

    Article  MATH  MathSciNet  Google Scholar 

  • Shorack G., Wellner J. (1986) Empirical processes with applications to statistics. Wiley, New York

    MATH  Google Scholar 

  • Xiang X. (1995a) A Berry-Esseen theorem for the kernel quantile estimator with application to studying the deficiency of qunatile estimators. Annals of the Institute of Statistical Mathematics 47(2): 237–251

    Article  MATH  MathSciNet  Google Scholar 

  • Xiang X. (1995b) Defficiency of the sample quantile estimator with respect to kernel quantile estimators for censored data. The Annals of Statistics 23(3): 836–854

    Article  MATH  MathSciNet  Google Scholar 

  • Xiang X., Vos P. (1997) Quantile estimators and covering probabilities. Journal of Nonparametric Statistics 7: 349–363

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Mathematics, Kyushu University, Fukuoka, Japan

    Yoshihiko Maesono

  2. Department of Statistics, School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia

    Spiridon Penev

Authors
  1. Yoshihiko Maesono
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Spiridon Penev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshihiko Maesono.

About this article

Cite this article

Maesono, Y., Penev, S. Edgeworth expansion for the kernel quantile estimator. Ann Inst Stat Math 63, 617–644 (2011). https://doi.org/10.1007/s10463-009-0241-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-009-0241-5

Keywords

Search

Navigation