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

Abstract

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 Hölder 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.

Appendix

Here we present the Bernstein inequality used in this study.

Bernstein’s inequality [11]. Let $X_{1},\ldots ,X_n$ be a sequence of independent random variables such that $\textbf{E}(X_{i})=0,\;\textbf{E}(X^2_{i})=\sigma ^2$ and $|X_i|\le M'<\infty$ ($i=1,\ldots ,n$). Then for each $\vartheta >0,$ we have

$$\begin{aligned} \mathbb {P}\left( \left| \frac{1}{n}\sum \limits _{i=1}^{n}X_i\right| >\vartheta \right) \le 2\exp \left( -\frac{n\vartheta ^2}{2\left( \sigma ^2+\frac{\vartheta M'}{3}\right) }\right) . \end{aligned}$$