Intensionally Biased Bootstrap Method

Abstract

A class of weighted bootstrap techniques, called biased bootstrap or b- bootstrap methods, is proposed. It is motivated by the need to adjust more conventional empirical methods, such as the "uniform" bootstrap, in a surgical way so as to alter some of their features while leaving others unchanged. Depending on the nature of the adjustment, the b-bootstrap can be used to reduce bias, or reduce variance, or render some characteristic equal to a predetermined quantity. Examples of the latter application include a b- bootstrap approach to hypothesis testing in nonparametric contexts, where the b-bootstrap enables simulation "under the null hypothesis" even when the latter is false; and a b-bootstrap competitor to Tibshirani's variance stabilisation method. An example of the bias-reduction application is adjustment of Nadaraya- Watson estimators of a regression mean, to make them competitive with local linear smoothing. Other applications include density estimation under constraints, outlier trimming, sensitivity analysis, skewness or kurtosis reduction and shrinkage.