On non-negative wavelet-based density estimators

Abstract

Standard wavelet-based density estimators may not retain some global properties of the curve, e.g. non-negativity and integral equal to one. This has the additional disadvantage in small samples of mass being taken out from “right places” and being put on “inappropriate places”, e.g. below the X-axis. We present a class of new wavelet-based estimation methods intended to retain asymptotic minimax optimality rates of standard methods by achieving non-negativity in a natural way. Moreover, the choice of the threshold level in the estimation process can be made in a simple adaptive manner. Basic for our procedure is the presentation of the density f (x) as a trace of an appropriate multivariate function expanded in a wavelet series. First, a new non-linear approximation of f(x) is proposed. The empirical version of the approximation yields the estimator. The estimators of the wavelet coefficients are also of non-linear type.