Isaac Scientific Publishing

Journal of Advanced Statistics

On the Performance of Confidence Intervals for Quantiles

Download PDF (232.2 KB) PP. 171 - 180 Pub. Date: September 1, 2016

DOI: 10.22606/jas.2016.13006


  • Yijun Zuo*
    Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA


Woodruff confidence interval for quantiles is a classical procedure and prevailing in practices and regarded as optimal one for many practitioners. This manuscript examines the performance of bootstrap based confidence interval and the classical Woodruff one for quantiles. It is found that the bootstrap procedure can outperform the Woodruff one in terms of coverage probability( accuracy) and the length of the intervals(efficiency). The validity of these theoretical findings for large sample is further confirmed in finite sample simulation studies.


Quantile, bootstrap, Bahadur representation, Confidence interval, Coverage probability, Length of confidence interval.


[1] R. R. Bahadur, “A note on quantiles in large samples," Annal of Mathematical Statistics, vol. 37, pp. 577-580, 1966.

[2] P. J. Bickel and D. A. Freedman, “Some asymptotic theory for the bootstrap," Annal of Statistics , vol. 9, pp. 1196-1217, 1981.

[3] T. J. DiCiccio and B. Efron, “Bootstrap Confidence intervals," Statistical Science, vol. 11, no. 2, pp. 189-228, 1996.

[4] B. Efron, “Bootstrap methods: another look at the jackknife," Annal of Statistics, vol. 7, pp. 1-26, 1979.

[5] M. Falk and E. Kaufmann, “Coverage probabilities of bootstrap-confidence intervals for quantiles," Annal of Statistics, vol. 19, pp. 485-495, 1991.

[6] J. K. Ghosh, “A new proof of the Bahadur representation of quantiles and an application," Annal of Mathematical Statistics, vol 42, pp. 1957-1961, 1971.

[7] P. Hall, “Bahadur representations for uniform resampling and importance resampling, with applications to asymptotic relative efficiency," Annal of Statistics, vol. 19, pp. 1062-1072, 1991.

[8] P. Hall, The Bootstrap and Edgeworth Expansion. Springer, New York, 1992.

[9] J. Kiefer, “On Bahadur’s representation of sample quantiles," Annal of Mathematical Statistics, vol. 38, pp. 1328-1342, 1967.

[10] R. D. Reiss, “On the accuracy of the normal approximation for quantiles," Annal of Probability, vol. 2, pp. 741-744, 1974.

[11] R. J. Serfling, Approximation theorems of mathematical statistics, Wiley, New York, 1980.

[12] J. Shao and Y. Chen, “Bootstrapping sample quantiles based on complex survey data under hot deck imputation," Statist. Sinica, Vol. 8, No. 4, pp. 1071-1085, 1998

[13] K. Singh, “On the asymptotic accuracy of Efron’s boostrap," Annal of Statistics, vol. 9, pp. 1187-1195, 1981.

[14] N. V. Smirnov, (1952). “Limit Distributions for the Terms of a Variational Series," American Mathematical Sociaty, Translation, No. 67, 1952.

[15] R. S. Woodruff, “Confidence intervals for medians and other positive measures," Journal of American Statistical Association, vol. 47, pp. 635-646, 1952.

[16] Y. Zuo, “Bahadur reprsentations for botstrap quantiles," Metrika, vol. 78, pp. 597-610, 2015.