Smooth hyperbolic wavelet deconvolution with anisotropic structure

Abstract

This paper considers a deconvolution regression problem in a multivariate setting with anisotropic structure and constructs an estimator of the function of interest using the hyperbolic wavelet basis. The deconvolution structure assumed is an anisotropic version of the smooth type (either regular-smooth or super-smooth). The function of interest is assumed to belong to a Besov space with anisotropic smoothness. Global performances of the presented hyperbolic wavelet estimators is measured by obtaining upper bounds on convergence rates in the $\mathscr{L}^{p}$-risk with $1\le p\le 2$ and $1\le p<\infty $ in the regular-smooth and super-smooth cases respectively. The results are compared and contrasted with existing convergence results in the literature.