Anisotropic multivariate deconvolution using projection on the Laguerre basis

Abstract

We investigate adaptive density estimation in the additive model Z=X+Y, where X and Y are independent d-dimensional random vectors with non-negative coordinates. Our goal is to recover the density of X from independent observations of Z, assuming the density of Y is known. In the d=1 case, an estimation procedure using projection on the Laguerre basis has already been studied. We generalize this procedure in the multivariate case: we establish non-asymptotic upper bounds on the mean integrated squared error of the estimator and we derive convergence rates on anisotropic functional spaces. Moreover, we provide data-driven strategies for selecting the right projection space (for d=1, we improve the previous projection procedure). We illustrate these procedures on simulated data, and in dimension d=1 we compare our procedure with the previous adaptive projection procedure.