Probability Theory and Related Fields

, Volume 80, Issue 2, pp 261–268 | Cite as

Exact convergence rate of bootstrap quantile variance estimator

  • Peter Hall
  • Michael A. Martin


It is shown that the relative error of the bootstrap quantile variance estimator is of precise order n-1/4, when n denotes sample size. Likewise, the error of the bootstrap sparsity function estimator is of precise order n-1/4. Therefore as point estimators these estimators converge more slowly than the Bloch-Gastwirth estimator and kernel estimators, which typically have smaller error of order at most n-2/5.


Relative Error Stochastic Process Probability Theory Convergence Rate Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babu, G.J.: A note on bootstrapping the variance of sample quantile. Ann. Inst. Stat. Math. 38, 439–443 (1986)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bloch, D.A., Gastwirth, J.L.: On a simple estimate of the reciprocal of the density function. Ann. Math. Stat. 39, 1083–1085 (1968)MathSciNetGoogle Scholar
  3. 3.
    Chung, K.L.: A course in probability theory. New York: Academic Press 1974Google Scholar
  4. 4.
    Csörgő, M.: Quantile processes with statistical applications. Philadelphia: SIAM 1983Google Scholar
  5. 5.
    David, H.A.: Order statistics, 2nd edn. New York: Wiley 1981Google Scholar
  6. 6.
    David, F.N., Johnson, N.L.: Statistical treatment of censored data, I. Biometrika 41, 228–240 (1954)MathSciNetGoogle Scholar
  7. 7.
    Efron, B.: Bootstrap methods: another look at the jackknife. Ann. Stat. 7, 1–26 (1979)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Falk, M.: On the estimation of the quantile density function. Stat. Probab. Lett. 4 69–73 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Farrell, R.H.: On the lack of a uniformly consistent sequence of estimators of a density function in certain cases. Ann. Math. Stat. 38, 471–474 (1967)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Farrell, R.H.: On the best obtainable asymptotic rates of convergence in estimation of a density function at a point. Ann. Math. Stat. 43, 170–180 (1972)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Ghosh, M., Parr, W.C., Singh, K., Babu, G.J.: A note on bootstrapping the sample median. Ann. Stat. 12, 1130–1135 (1985)MathSciNetGoogle Scholar
  12. 12.
    Hall, P., Heyde, C.C.: Martingale limit theory and its application. New York: Academic Press 1980Google Scholar
  13. 13.
    Maritz, J.S., Jarrett, R.G.: A note on estimating the variance of the sample median. J. Am. Stat. Assoc. 73, 194–196 (1978)Google Scholar
  14. 14.
    McKean, J.W., Schrader, R.M.: A comparison of methods for studentizing the sample median. Comm. Statist. Ser. B, Simul. Computa. 13, 751–773 (1984)Google Scholar
  15. 15.
    Parzen, E.: Nonparametric statistical data modeling. J. Am. Stat. Assoc. 7, 105–131 (1979)MathSciNetGoogle Scholar
  16. 16.
    Pearson, K., Pearson, M.V.: On the mean character and variance of a ranked individual, and the mean and variance of the intervals between ranked individuals, I: symmetrical distributions (normal and rectangular). Biometrika 23, 364–397 (1931)MathSciNetGoogle Scholar
  17. 17.
    Pearson, K., Pearson, M.V.: On the mean character and variance of a ranked individual, and the mean and variance of the intervals between ranked individuals, II: case of certain skew curves. Biometrika 24, 203–279 (1932)MathSciNetGoogle Scholar
  18. 18.
    Rosenblatt, M.: Curve estimates. Ann. Math. Stat. 42, 1815–1842 (1971)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Sheather, S.J.: A finite sample estimate of the variance of the sample median. Stat. Probab. Lett. 4, 337–342 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Sheather, S.J.: An improved data-based algorithm for choosing the window width when estimating the density at a point. Comput. Stat. Data Anal. 4, 61–65 (1986)CrossRefGoogle Scholar
  21. 21.
    Sheather, S.J.: Assessing the accuracy of the sample median: estimated standard errors versus interpolated confidence intervals. In: Dodge, Y. (ed.) Statistical data analysis based on the L 1-norm, pp. 203–215. Amsterdam: North-Holland 1987Google Scholar
  22. 22.
    Sheather, S.J., Maritz, J.S.: An estimate of the asymptotic standard error of the sample median. Aust. J. Stat. 25, 109–122 (1983)MathSciNetGoogle Scholar
  23. 23.
    Wahba, G.: Optimal convergence properties of variable knot, kernel and orthogonal series methods for density estimation. Am. Stat. 3, 15–29 (1975)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Welsh, A.H.: Kernel estimates of the sparsity function. In: Dodge, Y. (ed.) Statistical data analysis based on the L 1-norm pp. 369–377. Amsterdam: North-Holland 1987Google Scholar
  25. 25.
    Welsh A.H.: Asymptotically efficient estimation of the sparsity function at a point. Stat. Probab. Lett., to appearGoogle Scholar
  26. 26.
    Van Zwet, W.R.: Convex, Transformations of Random Variables. Mathematical Centre Tracts No. 7. Amsterdam: Mathematisch Centrum 1964Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Peter Hall
    • 1
  • Michael A. Martin
    • 1
  1. 1.Department of StatisticsAustralian National UniversityCanberraAustralia

Personalised recommendations