Summary
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.
Article PDF
Similar content being viewed by others
References
Babu, G.J.: A note on bootstrapping the variance of sample quantile. Ann. Inst. Stat. Math. 38, 439–443 (1986)
Bloch, D.A., Gastwirth, J.L.: On a simple estimate of the reciprocal of the density function. Ann. Math. Stat. 39, 1083–1085 (1968)
Chung, K.L.: A course in probability theory. New York: Academic Press 1974
Csörgő, M.: Quantile processes with statistical applications. Philadelphia: SIAM 1983
David, H.A.: Order statistics, 2nd edn. New York: Wiley 1981
David, F.N., Johnson, N.L.: Statistical treatment of censored data, I. Biometrika 41, 228–240 (1954)
Efron, B.: Bootstrap methods: another look at the jackknife. Ann. Stat. 7, 1–26 (1979)
Falk, M.: On the estimation of the quantile density function. Stat. Probab. Lett. 4 69–73 (1986)
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)
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)
Ghosh, M., Parr, W.C., Singh, K., Babu, G.J.: A note on bootstrapping the sample median. Ann. Stat. 12, 1130–1135 (1985)
Hall, P., Heyde, C.C.: Martingale limit theory and its application. New York: Academic Press 1980
Maritz, J.S., Jarrett, R.G.: A note on estimating the variance of the sample median. J. Am. Stat. Assoc. 73, 194–196 (1978)
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)
Parzen, E.: Nonparametric statistical data modeling. J. Am. Stat. Assoc. 7, 105–131 (1979)
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)
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)
Rosenblatt, M.: Curve estimates. Ann. Math. Stat. 42, 1815–1842 (1971)
Sheather, S.J.: A finite sample estimate of the variance of the sample median. Stat. Probab. Lett. 4, 337–342 (1986)
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)
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 1987
Sheather, S.J., Maritz, J.S.: An estimate of the asymptotic standard error of the sample median. Aust. J. Stat. 25, 109–122 (1983)
Wahba, G.: Optimal convergence properties of variable knot, kernel and orthogonal series methods for density estimation. Am. Stat. 3, 15–29 (1975)
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 1987
Welsh A.H.: Asymptotically efficient estimation of the sparsity function at a point. Stat. Probab. Lett., to appear
Van Zwet, W.R.: Convex, Transformations of Random Variables. Mathematical Centre Tracts No. 7. Amsterdam: Mathematisch Centrum 1964
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hall, P., Martin, M.A. Exact convergence rate of bootstrap quantile variance estimator. Probab. Th. Rel. Fields 80, 261–268 (1988). https://doi.org/10.1007/BF00356105
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00356105