Statistica Sinica 4(1994), 477-492

ON EXACTNESS OF THE PARAMETRIC DOUBLE

BOOTSTRAP

F. W. Scholz

Boeing Computer Services

Dedicated to Erich L. Lehmann
on the occasion of his 75th birthday

Abstract: Given a data set X~Pψ,η and estimates and we are interested in confidence bounds for the real parameter ψ. Let and assume that is a pivot with pivot distribution H. 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 of D or H, 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.