Anisotropic functional deconvolution with long-memory noise: the case of a multi-parameter fractional Wiener sheet

Abstract

We look into the minimax results for the anisotropic two-dimensional functional deconvolution model with the two-parameter fractional Gaussian noise. We derive the lower bounds for the L p -risk, 1 ≤ p < ∞, and taking advantage of the Riesz poly-potential, we apply a wavelet-vaguelette expansion to de-correlate the anisotropic fractional Gaussian noise. We construct an adaptive wavelet hard-thresholding estimator that attains asymptotically quasioptimal convergence rates in a wide range of Besov balls. Such convergence rates depend on a delicate balance between the parameters of the Besov balls, the degree of ill-posedness of the convolution operator and the parameters of the fractional Gaussian noise. A limited simulations study confirms theoretical claims of the paper. The proposed approach is extended to the general r-dimensional case, with r > 2, and the corresponding convergence rates do not suffer from the curse of dimensionality.

Adaptive estimation for the nonparametric bivariate additive model in random design with long-memory dependent errors

“…4. To compute the hard thresholding estimators (10) and (11), we use the same Daubechies wavelet with 3 vanishing moments, where the thresholds are given in (21). And highest resolution levels J 1 and J 2 are given by (32).…”

“…Thus, the estimator for the bivariate function U (t, x) is the sum of its components' own estimators (12), (10) and (11), as follows…”

Estimation in nonparametric regression model with additive and multiplicative noise via Laguerre series

“…Remark 9 Notice that the i.i.d. case can also be handled using estimators (5) along with the choice of thresholds (11) and truncation level M based on (12) and achieve the same convergence rates.…”