Wavelet Optimal Estimations for a Multivariate Probability Density Function Under Weighted Distribution

The purpose of this paper is the pointwise estimation of a multivariate probability density function with weighted distribution using wavelet methods. New theoretical contributions are provided; Point-wise convergence rates of wavelet estimators are established in the local Holder space. First, a lower bound is provided for all the possible estimators. Specially, a linear wavelet estimator is defined and turned out to be the optimal one in the considered setting. Subsequently, regarding the adaptive estimation problem, a nonlinear estimator is proposed as usual and discussed. Finally, a new data driven wavelet estimator is introduced and shown to be completely adaptive and almost optimal.