Formulae for Mean Integrated Squared Error of Nonlinear Wavelet-Based Density Estimators

Abstract

We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based density estimators. We show that, unlike the analogous situation for kernel density estimators, this MISE formula is relatively unaffected by assumptions of continuity. In particular, it is available for densities which are smooth in only a piecewise sense. Another difference is that in the wavelet case the classical MISE formula is valid only for sufficiently small values of the bandwidth. For larger bandwidths MISE assumes a very different form and hardly varies at all with changing bandwidth. This remarkable property guarantees a high level of robustness against oversmoothing, not encountered in the context of kernel methods. We also use the MISE formula to describe an asymptotically optimal empirical bandwidth selection rule.