The Annals of Statistics

Breakdown theory for bootstrap quantiles

Kesar Singh

Full-text: Open access


A general formula for computing the breakdown point in robustness for the $t$th bootstrap quantile of a statistic $T_n$ is obtained. The answer depends on $t$ and the breakdown point of $T_n$. Since the bootstrap quantiles are vital ingredients of bootstrap confidence intervals, the theory has implications pertaining to robustness of bootstrap confidence intervals. For certain $L$ and $M$ estimators, a robustification of bootstrap is suggested via the notion of Winsorization.

Article information

Ann. Statist., Volume 26, Number 5 (1998), 1719-1732.

First available in Project Euclid: 21 June 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G09: Resampling methods 62G15

Bootstrap quantiles breakdown in robustness $L$ and $M$ estimators Winsorization.


Singh, Kesar. Breakdown theory for bootstrap quantiles. Ann. Statist. 26 (1998), no. 5, 1719--1732. doi:10.1214/aos/1024691354.

Export citation


  • CARROLL, R. J. 1978. On almost sure expansion for M-estimators. Ann. Statist. 6 314 318. Z.
  • DVORETZKY, A., KIEFER, J. C. and WOLFOWITZ, J. 1956. Asy mptotic minimax character of the sample distribution function and the classical multinomial estimator. Ann. Math. Statist. 27 642 669. Z.
  • HE, X., SIMPSON, D. G. and PORTNOY, S. L. 1990. Breakdown resistance of tests. J. Amer. Statist. Assoc. 85 446 452. Z.
  • HUBER, P. J. 1964. Robust estimation of a location parameter. Ann. Math. Statist. 35 73 101. Z.
  • HUBER, P. J. 1981. Robust Statistics. Wiley, New York. Z. Z
  • LEHMANN, E. L. 1983. Theory of Point Estimation. Wiley, New York transferred to Wadsworth,. 1991. Z.
  • LIU, R. and SINGH, K. 1993. A quality index based on data-depth. J. Amer. Statist. Assoc. 88 252 260. Z.
  • MASSART, P. 1990. The tight constant in the Dvoretzky, Kiefer, Wolfowitz inequality. Ann. Probab. 18 1269 1283. Z.
  • STROMBERG, A. J. 1997. Robust covariance estimates based on resampling. J. Statist. Plann. Inference 57 321 334. Z.
  • YLVISAKER, D. 1977. Test resistance. J. Amer. Statist. Assoc. 72 551 556.