Wavelet Density Estimation for Weigthed Data.

We consider the estimation of a density function on the basis of a random sample from a weighted distribution. We propose linear and nonlinear wavelet density estimators, and provide their asymptotic formulae for mean integrated squarer error. In particular, we derive an analogue of the asymptotic formula of the mean integrated square error in the context of kernel density estimators for weighted data, admitting an expansion with distinct squared bias and variance components. For nonlinear wavelet density estimators, unlike the analogous situation for kernel or linear wavelet density estimators, this asymptotic formula of the mean integrated square error is relatively unaffected by assumptions of continuity, and it is available for densities which are smooth in only a piecewise sense. Comparisons with two other methods proposed in the literature, namely with a kernel density estimator and with a Fourier series density estimator, are also given.