Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞

Abstract

We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the L p -risk (1 ≤ p < ∞). Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.

Minimax lower bounds for the simultaneous wavelet deconvolution with fractional Gaussian noise and unknown kernels

“…(iii) For δ = 0, our rates coincide, up to some logarithmic factor of ε ≍ n −1/2 , with the upper bounds obtained in Kulik et al (2015) in the regular-smooth convolution case.…”

“…To determine the choices of J, m 0 and λ α j;ε,δ in ( 9) and (10), it is necessary to evaluate the variance of (7). Thus, recall that by (8), one has |m| ≍ 2 j , and define for some constant 0 < ρ < 1/2, the sets Ω 1 and Ω 2 as…”

“…The case δ = 0 and α 1l = 1, l = 1, 2, · · · , M , corresponds to the simultaneous deconvolution with white noise and known kernel studied in De Canditiis and Pensky (2006), while the case δ = 0 corresponds to the multichannel deconvolution model with long-range dependence and known kernel investigated in Kulik et al (2015). In addition, the case M = 1 and α 2l = 1, pertains to the one channel blind deconvolution model with fGn investigated in Benhaddou (2018a), while the case δ = 0 and M = 1, corresponds to the one channel deconvolution model with fBm and known kernel discussed in Wishart (2013).…”

Laplace deconvolution with dependent errors: a minimax study

“…More specifically, the stronger the LM is, the slower the convergence rates will be, compared to Benhaddou et al (2013). This detrimental effect of LM on convergence rates was pointed out in Wishart (2013), Kulik et al (2015), Benhaddou (2016) and Benhaddou (2017b). (v) Note that our rates of convergence are not directly comparable to those in Kulik et al (2015), since their rates pertain to an estimator of a one-dimensional function using a finite number of (M different) channels, while in the present work the convergence rates are associated with the estimation of a two-dimensional function and that the number of profiles M is asymptotic.…”

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

“…Theorem 2 Let f (., .) be the wavelet estimator in (14), with λ(j 1 , j 2 ) given by (19) or (20) and, J 1 and J 2 given by (21). Let min{s 1 , s 2 } ≥ max{ 1 p , 1 2 }, and let conditions (3), ( 9), ( 10) and ( 25) hold.…”