Adaptive bandwidth choice

Abstract

In this paper, we consider the problem of bandwidth choice in the parallel settings of nonparametric kernel smoothed spectra] density and probability density estimation. We propose a new class of 'plug-in' type bandwidth estimators, and show their favorable asymptotic properties. The new estimators automatically adapt to the degree of underlying smoothness which is unknown. The main idea behind the new estimators is the use of infinite-order 'flat-top' kernels for estimation of the constants implicit in the formulas giving the asymptotically optimal bandwidth choices. The proposed bandwidth choice rule for 'flat-top' kernels has a direct analogy with the notion of thresholding in wavelets. It is shown that the use of infinite-order kernels in the pilot estimator has a twofold advantage: (a) accurate estimation of the bandwidth constants, and (b) easy determination of the required 'pilot' kernel bandwidth.