Canadian Journal of Statistics

Do bootstrap confidence procedures behave well uniformly in P ?

Journal Article

Abstract

An important statistical problem is to construct a confidence set for some functional T(P) of some unknown probability distribution P. Typically, this involves approximating the sampling distribution Jn(P) of some pivot based on a sample of size n from P. A bootstrap procedure is to estimate Jn(P) by Jn(&Pcirc;n), where P̂n is the empirical measure based on a sample of size n from P. Typically, one has that Jn(P) and Jn(P̂n) are close in an appropriate sense. Two questions are addressed in this note. Are Jn(P) and Jn(P̂n) uniformly close as P varies as well? If so, do confidence statements about T(P) possess a corresponding uniformity property? In the case T(P) = P, the answer to the first questions is yes; the answer to the second is no. However, bootstrap confidence statements about T(P) can be made uniform over a restricted, though large, class of P. Similar results apply to other functional T(P).

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