Statistica Sinica

F. W. Scholz

Dedicated to Erich L. Lehmann

on the occasion of his 75th birthday

Abstract:Given a data setX~P_{ψ,η }and estimates and we are interested in confidence bounds for the real parameter ψ. Let and assume that is a pivot with pivot distributionH. Assume that is nondecreasing in ψ for fixed and . Then it is possible to construct exact, transformation equivariant confidence bounds for ψ. It is shown that a modified double bootstrap procedure yields exactly these bounds without knowledge ofDorH, provided the number of bootstrap samples becomes infinite. Although the existence of exact pivots is special, it is plausible that the proposed method will yield approximate confidence bounds, when there are approximate local pivots. This aspect is explored analytically and by simulation in two examples.

Key words and phrases:Bootstrap, nonparametric bootstrap, confidence bounds, percentile methods, percentile-t, pivot, prepivoting, root, calibrated confidence sets, Behrens-Fisher problem.