Anisotropic functional deconvolution for the irregular design: A minimax study

Abstract

Anisotropic functional deconvolution model is investigated in the bivariate case when the design points ti , and xl , are irregular and follow known densities h 1, h 2, respectively. In particular, we focus on the case when the densities h 1 and h 2 have singularities, but and are still integrable on [0, 1]. We construct an adaptive wavelet estimator that attains asymptotically near-optimal convergence rates in a wide range of Besov balls. The convergence rates are completely new and depend on a balance between the smoothness and the spatial homogeneity of the unknown function f, the degree of ill-posed-ness of the convolution operator and the degrees of spatial irregularity associated with h 1 and h 2. Nevertheless, the spatial irregularity affects convergence rates only when f is spatially inhomogeneous in either direction.