Wavelet Estimation for Density and Copula Functions

This article investigates the problem of univariate and bivariate density estimation using wavelet decomposition techniques. Special attention is given to the estimation of copula functions, which capture the dependence structure between random variables independent of their marginals. We consider two distinct frameworks: the case of independent and identically distributed (i.i.d.) variables and the case where variables are dependent, allowing us to highlight the impact of the dependence structure on the performance of wavelet-based estimators. Building on this framework, we propose a novel iterative thresholding method applied to the detail coefficients of the wavelet transform. This iterative scheme aims to enhance noise reduction while preserving significant structural features of the underlying density or copula function. Numerical experiments illustrate the effectiveness of the proposed method in both univariate and bivariate settings, particularly in capturing localized features and discontinuities in the presence of varying dependence patterns.