Abstract
The bootstrap method was used to reduce the sample volume at estimating the quantile function. Accuracy of the quantile sample estimate vs. the distribution of random variable was established analytically. A numerical example of calculation of the quantiles using the proposed bootstrap procedure for the uniform, normal, and Cauchy distributions was considered. An approximate formula for calculation of the quantile of the normal distribution was determined.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Efron, B., Bootstrap Methods: Another Look at Jackknife, Ann. Statist., 1979, vol. 7, no. 1, pp. 1–25.
Quenouille, M.H., Approximate Tests of Correlation in Time Series, J. Roy. Statist. Soc. Ser. B, 1949, pp. 68–84.
Efron, B., Netraditsionnye metody mnogomernogo statisticheskogo analiza. Sbornik statei (Nontraditional Methods of Multidimensional Statistical Analysis. Collected Papers), Moscow: Finansy i Statistika, 1988.
Beran, R. and Ducharme, G.R., Asymptotic Theory for Bootstrap Methods in Statistic, Les Publications CRM, Centre de rechereches mathematiques, 1991.
Davison, A.C. and Hinkley, D.V., Bootstrap Methods and Their Application, Cambridge: Cambridge Univ. Press, 1997.
Hall, P., The Bootstrap and Edgeworth Expansion, New York: Springer, 1992.
Horowitz, J.L., The Bootstrap. Handbook of Economertics, 1999, vol. 5.
Hall, P., Methodology and Theory for the Bootstrap, in Handbook of Econometrics, Engle, R.F. and McFadden, D.F., Eds., 1994, vol. 4, Ch. 39.
Horowitz, J.L., Bootstrap Methods in Econometrics: Theory and Numerical Perfomance, in Advance in Economics and Econometrics: Theory and Applications. Seventh World Congress, Kreps, D.M. and Wallis, K.F., Eds., 1997, vol. 3, Ch. 7.
Kibzun, A.I. and Kan, Yu.S., Stochastic Programming Problems with Probability and Quantile Functions, Chichester: Wiley, 1996.
Ivchenko, G.I. and Medvedev, Yu.I., Matematicheskaya statistika (Mathematical Statistics), Moscow: Vysshaya Shkola, 1984.
David, H., Order Statistics, New York: Wiley, 1970. Translated under the title Poryadkovye statistiki, Moscow: Nauka, 1979.
Galambos, J., The Asymptotic Theory of Extreme Order Statistics, New York: Wiley, 1978. Translated under the title Asimptoticheskaya teoriya exstremal’hykh poryadkovykh statistik (Asymptotic Theory of Extremal-Order Statistics), Moscow: Nauka, 1984.
Dwight, H.B., Tables of Integrals and Other Mathematical Data, New York: Macmillan, 1961. Translated under the title Tablitsy integralov i drugie matematicheskie formuly, Moscow: Nauka, 1978.
Ruben, H., A New Asymptotic Expansion for the Normal Probability Integral and Mill’s Ratio, J. Royal Statist. Soc. Ser. B (Methodological), 1962, vol. 24, no. 1, pp. 177–179.
Corless, R.M., Gonnet, G.H., Hare, D.E.G., et al., On the Lambert W Function, Advances Comput. Math., 1996, vol. 5, pp. 329–359.
Ho, Y. and Lee, S., Iterated Smoothed Bootstrap Confidence Intervals for Population Quantiles, Annals. Statis., 2005, vol. 33, no. 1, pp. 437–462.
Hall, P., Horowitz, J.L., and Jing, B.-Y., On Blocking Rules for the Bootstrap with Depended Data, Biometrika, 1995, no. 85, pp. 561–574.
Author information
Authors and Affiliations
Additional information
Original Russian Text © B.V. Vishnyakov, A.I. Kibzun, 2007, published in Avtomatika i Telemekhanika, 2007, No. 11, pp. 46–60.
This work was supported by the Russian Foundation for Basic Research, project no. 05-08-17963.
Rights and permissions
About this article
Cite this article
Vishnyakov, B.V., Kibzun, A.I. Application of the bootstrap method for estimation of the quantile function. Autom Remote Control 68, 1931–1944 (2007). https://doi.org/10.1134/S0005117907110045
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0005117907110045