Wavelet density estimation for weighted data

• We consider the problem of the estimation of the density function from weighted data.
• We propose a novel nonparametric, linear and nonlinear, wavelet density estimator for weighted data.
• We provide the asymptotic formulae for MISE, admitting an expansion with distinct squared bias and variance components.
• We show that this asymptotic formula of the mean integrated square error is relatively unaffected by assumptions of continuity.
• We point out that nonlinear wavelet estimators possess a property which guarantees a high level of robustness against oversmoothing.

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 squared 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 only in a piecewise sense. We illustrate the behavior of the proposed linear and nonlinear wavelet density estimators in finite sample situations both in simulations and on a real-life dataset. Comparisons with a kernel density estimator are also given.