The Annals of Statistics

A Note on Bootstrapping the Sample Median

Malay Ghosh, William C. Parr, Kesar Singh, and G. Jogesh Babu

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Efron (1979, 1982), in his treatment of the bootstrap, discusses its use for estimation of the asymptotic variance of the sample median, in the sampling situation of independent and identically distributed random variables with common distribution function $F$ having a positive derivative continuous in a neighborhood of the true median $\mu$. The natural conjecture that the bootstrap variance estimator converges almost surely to the asymptotic variance is shown by an example to be false unless a tail condition is imposed on $F$. We prove that such strong convergence does hold under the fairly nonrestrictive condition that $E\lbrack\mid X^\alpha\rbrack < \infty$ for some $\alpha > 0$.

Article information

Ann. Statist., Volume 12, Number 3 (1984), 1130-1135.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 62G05: Estimation

Bootstrap median standard error estimation


Ghosh, Malay; Parr, William C.; Singh, Kesar; Babu, G. Jogesh. A Note on Bootstrapping the Sample Median. Ann. Statist. 12 (1984), no. 3, 1130--1135. doi:10.1214/aos/1176346731.

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