PREVENTING THE DIRAC DISASTER: Wavelet Based Density Estimation

This paper addresses the problem of choosing the optimal number of basis functions in constructing wavelet series density estimators. It is well known that projection estimators tend to overrt the density if the number of basis functions in the orthogonal expansion is too large. In extreme cases the estimator is close to the Dirac function concentrated at the observations. We propose a roughness measure of wavelet estimators and establish a data driven method for determining the number of levels to be included in the estimate. Our method exploits the idea of Good and Gaskins (1971) who used the Fisher information functional as a roughness penalty measure. The method is demonstrated on simulated data.