Reducing variability using bootstrap methods with qualitative constraints

Abstract

Parametric methods are often attractive for small to moderate samples, since, relative to nonparametric techniques, they reduce variability of a wide range of statistical procedures. However, when sample size is small there is often little empirical evidence to support a particular model, and so, while variance can be reduced, bias may be increased. In theory, smoothing can be used to reduce variability of nonparametric procedures, but the difficulty of choosing the smoothing parameter can be a serious drawback. In the present paper we propose an alternative approach. We suggest implicitly smoothing nonparametric distribution estimates by enforcing the same sort of qualitative constraint that parametric methods attempt to reflect. We show theoretically that such a method produces the same order of variance reduction as explicit smoothing, even when the smoothing parameter for the latter is chosen optimally. Furthermore, we demonstrate numerically that imposing qualitative constraints on distribution estimates, and on the nonparametric bootstrap, does in fact produce important reductions in variability for small sample sizes.