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Biometrika 1987 74(3):469-479; doi:10.1093/biomet/74.3.469
© 1987 by Biometrika Trust
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The bootstrap: To smooth or not to smooth?

B. W. SILVERMAN and G. A. YOUNG

School of Mathematical Sciences, University of Bath Bath BA2 7AY, U.K.
Statistical Laboratory, University of Cambridge Cambridge CB2 1SB, U.K.

The bootstrap and smoothed bootstrap are considered as alternative methods of estimating properties of unknown distributions such as the sampling error of parameter estimates. Criteria are developed for determining whether it is advantageous to use the smoothed bootstrap rather than the standard bootstrap. Key steps in the argument leading to these criteria include the study of the estimation of linear functionals of distributions and the approximation of general functionals by linear functionals. Consideration of an example, the estimation of the standard error in the variance-stabilized sample correlation coefficient, elucidates previously-published simulation results and also illustrates the use of computer algebraic manipulation as a useful technique in asymptotic statistics. Finally, the various approximations used are vindicated by a simulation study.

Key Words: Bootstrap • Computer algebra • Density estimation • Kernel • Resampling • Smoothed bootstrap


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