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A note on bootstrapping the sample median. (English) Zbl 0541.62010

Summary: B. Efron [ibid. 7, 1-26 (1979; Zbl 0406.62024) and CBMS-NSF Reg. Conf. Ser. Appl. Math. 38 (1982; Zbl 0496.62036)], 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 \(E[| X^{\alpha}|]<\infty\) for some \(\alpha>0\).

MSC:

62E20 Asymptotic distribution theory in statistics
62G05 Nonparametric estimation
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