## The Annals of Statistics

### A Note on Bootstrapping the Sample Median

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346731

Digital Object Identifier
doi:10.1214/aos/1176346731

Mathematical Reviews number (MathSciNet)
MR751302

Zentralblatt MATH identifier
0541.62010

JSTOR