Summary
The normalized Kolmogorov-Smirnov and variational distance between the distribution of the sampleq-quantile and the pertaining smooth bootstrap distribution are asymptotically distributed like the absolute value of a normal random variable. The distribution functions of these random distances may serve as measures of the accuracy of the bootstrap procedure.
It is shown that these distribution functions of random distances, and thus the accuracy of the bootstrap procedure, can again consistently be estimated by means of the bootstrap technique.
References
Beran, R.J.: Jackknife approximations to bootstrap estimates. Ann. Stat.12, 101–118 (1984)
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)
Falk, M.: Weak convergence of the maximum error of the bootstrap quantile estimate. Preprint, University of Siegen 1987
Falk, M., Reiss, R.-D.: Weak convergence of smoothed and nonsmoothed bootstrap quantile estimates. Ann. Probab.17, 362–371 (1989)
Hall, P., Marron, J.S.: On the amount of noice inherent in bandwidth selection for a kernel density estimate. Ann. Stat.15, 163–181 (1987)
Marron, J.S.: A comparison of cross validation techniques in density estimation. Ann. Stat.15, 152–162 (1987)
Reiss, R.-D.: Asymptotic expansions for sample quantiles. Ann. Probab.4, 249–258 (1976)
Reiss, R.-D.: Asymptotic independence of distributions of normalized order statistics of the underlying probability measure. J. Multivariate Anal.11, 386–399 (1981)
Singh, K.: On the asymptotic accuracy of Efron's bootstrap. Ann. Stat.9, 1187–1195 (1981)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Falk, M., Reiss, R.D. Bootstrapping the distance between smooth bootstrap and sample quantile distribution. Probab. Th. Rel. Fields 82, 177–186 (1989). https://doi.org/10.1007/BF00354758
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00354758
Keywords
- Distribution Function
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Variational Distance