On efficient bootstrap simulation

Abstract

It is shown that three new, efficient bootstrap algorithms are asymptotically equivalent. This is done in two ways. First, asymptotic formulae for variances and mean squared errors are derived, and shown to be identical. Secondly, it is demonstrated that two of the methods may be viewed as approximations to the third. The three algorithms considered are the balanced bootstrap and the linear approximation method proposed by Davison, Hinkley & Schechtman (1986), and a centring method proposed by Efron in the context of bias estimation. It is shown that each reduces the order of magnitude of mean squared error by the factor n −1 , where n is sample size. These results apply to smooth functions of means.