Anisotropic Functional Deconvolution for the irregular design with dependent long-memory errors

TL;DR

This paper develops an adaptive wavelet estimator for anisotropic functional deconvolution with irregular design points and dependent long-memory errors, achieving near-optimal convergence rates that depend on smoothness, ill-posedness, and long-memory parameters.

Contribution

It introduces a novel estimator that handles irregular design densities with singularities and long-memory errors, providing new convergence rate results in this complex setting.

Findings

Convergence rates depend on smoothness, ill-posedness, and long-memory parameters.

Estimator attains near-optimal rates under Gaussian and sub-Gaussian errors.

Spatial irregularity influences convergence only for spatially inhomogeneous functions.

Abstract

Anisotropic functional deconvolution model is investigated in the bivariate case under long-memory errors when the design points $t_i$, $i=1, 2, \cdots, N$, and $x_l$, $l=1, 2, \cdots, M$, 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.