Anisotropic Functional Deconvolution for the irregular design with dependent long-memory errors
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Anisotropic functional deconvolution model is investigated in the bivariate case under long-memory errors when the design points t_i, i=1, 2, ..., N, and x_l, l=1, 2, ..., M, 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 1/h_1 and 1/h_2 are still integrable on [0, 1]. Under both Gaussian and sub-Gaussian errors, we construct an adaptive wavelet estimator that attains asymptotically near-optimal convergence rates that deteriorate as long-memory strengthens. 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, the long-memory parameter in addition to 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.
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