Reducing variability using bootstrap methods with qualitative constraints

Parametric methods are often attractive for small to moderate samples, sinc e, relative to nonparametric techniques, they reduce variability of a wide range of statistical procedures. However, when sample size is small there i s often little empirical evidence to support a particular model, and so, wh ile variance can be reduced,bias may be increased. In theory, smoothing can be used to reduce variability of nonparametric procedures, but the difficu lty of choosing the smoothing parameter can be a serious drawback. In the p resent paper we propose an alternative approach. We suggest implicitly smoo thing nonparametric distribution estimates by enforcing the same sort of qu alitative constraint that parametric methods attempt to reflect. We show th eoretically that such a method produces the same order of variance reductio n as explicit smoothing, even when the smoothing parameter for the latter i s chosen optimally. Furthermore, we demonstrate numerically that imposing q ualitative constraints on distribution estimates, and on the nonparametric bootstrap, does in fact produce important reductions in variability for sma ll sample sizes.