Department of Statistics, Texas A&M University, College Station, Texas 77843-3143, U.S.A.

(Received October 7, 1993; revised September 30, 1994)

Abstract.    In this paper we develop the technique of a generalized rescaling in the smoothed bootstrap, extending Silverman and Young's idea of shrinking. Unlike most existing methods of smoothing, with a proper choice of the rescaling parameter the rescaled smoothed bootstrap method produces estimators that have the asymptotic minimum mean (integrated) squared error, asymptotically improving existing bootstrap methods, both smoothed and unsmoothed. In fact, the new method includes existing smoothed bootstrap methods as special cases. This unified approach is investigated in the problems of estimation of global and local functionals and kernel density estimation. The emphasis of this investigation is on theoretical improvements which in some cases offer practical potential.

Key words and phrases:    Bootstrap, functional estimation, kernel density estimation, mean integrated squared error, mean squared error, quantile, rescaling, smoothing.

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