Multichannel deconvolution with long-range dependence: A minimax study

Abstract

We consider the problem of estimating the unknown response function in the multichannel deconvolution model with long-range dependent Gaussian errors. We do not limit our consideration to a specific type of long-range dependence rather we assume that the errors should satisfy a general assumption in terms of the smallest and larger eigenvalues of their covariance matrices. We derive minimax lower bounds for the quadratic risk in the proposed multichannel deconvolution model when the response function is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of the response function that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls. It is shown that the optimal convergence rates depend on the balance between the smoothness parameter of the response function, the kernel parameters of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the total number of channels. Some examples of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation are used to illustrate the application of the theory we developed. The optimal convergence rates and the adaptive estimators we consider extend the ones studied by Sapatinas (2009, 2010) for independent and identically distributed Gaussian errors to the case of long-range dependent Gaussian errors.

Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞Kulik1, Sapatinas2, Wishart3 2015Applied and Computational Harmonic AnalysisSelf Cite12380View full textAdd to dashboardCite

“…A possible direction for future research is to explore and extend our upper bounds to minimax type rates towards the direction of [21] obtained for the L 2 -risk in the discrete model when the number of channels, M , also depend on the total number of observations n, i.e., M = M n .…”

“…where c n is given by (21), since it is easily seen in this case that τ 2 j = O(2 2jν * ) with ν * given by (6) (compare also with p. 306 of [3]). A similar bound can be shown for the box-car case with ν * replaced with ν * given by (8).…”

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

“…Lemma 2 Let βω be defined in (8) and let s 10 > max{ν, r}. Then, under the conditions (3), ( 9) and ( 10), as M, N → ∞, simultaneously, one has…”

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

“…Long-memory has been investigated quite considerably in the standard deconvolution model. One can list; Wang (1996, Wishart (2013), Benhaddou et al (2014) and Kulik et al (2015). In a few other relevant contexts, LM was also investigated in density deconvolution in , and in the Laplace deconvolution in Benhaddou (2018) where the unknown response function f (·) is non-periodic and defined on the entire positive real half-line.…”