Richardson Extrapolation and the Bootstrap
نویسندگان
چکیده
Simulation methods, in particular Efron's (1979) bootstrap, are being applied more and more widely in statistical inference. Given data, (X1,* ,Xn), distributed according to P belonging to a hypothesized model P the basic goal is to estimate the distribution Lp of a function Tn (X1, * *Xn,P). The bootstrap presupposes the existence of an estimate P (X1, Xn) and consists of estimating Lp by the distribution L* of Tn(XI,* ,Xn,P) where (X1, * * ,Xn ) is distributed according to P. The method is particularly of interest when L*, though known in principle, is realistically only computable by simulation. Such computation can be expensive if n is large and Tn is very complex see for instance the multivariate goodness of fit tests of Beran and Millar (1985). Even when application of the bootstrap to a single data set is not excessively expensive, Monte Carlo studies of the bootstrap are another matter. We propose a method based on the classical ideas of Richardson extrapolation for reducing the computational cost inherent in bootstrap simulations and Monte Carlo studies of the bootstrap by doing the simulations for statistics based on two smaller sample sizes.
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