Results from MathSciNet: Mathematical Reviews on the Webc Copyright American Mathematical Society 2023
MR4545120 62G05 62G07 62G20
Capitao-Miniconi, Jérémie (F-UPS7-LM) ; Gassiat, Elisabeth (F-UPS7-LM)
Deconvolution of spherical data corrupted with unknown noise. (English.
English summary)Electron. J. Stat. 17 (2023), no. 1, 607–649.

In this article, the authors study a deconvolution problem of multivariate spherical data.
Suppose that an observation Y is contaminated in such a way that Y = X + , wherethe signal X is a random variable that takes values on a (d − 1)-dimensional sphere (ford ≥ 2) with an unknown center and an unknown radius, and the noise  ∈ Rd is anotherrandom variable that is independent of the signal. Both signal and noise distributions
are unknown, and the only condition imposed is that d coordinates of the noise areindependently distributed.
As a natural extension of the results in [É. Gassiat, S. Le Corff and L. Lehéricy, Ann.
Statist. 50 (2022), no. 1, 303–323; MR4382018], the authors develop estimation methodsof the center, radius and density of the signal on the sphere using n iid observations{Yi }ni=1 , with the noise distribution left unspecified. In the first step of their analysis,
it is demonstrated that the center, radius and density of the signal on the sphere are
identifiable if the noise distribution is centered and has a finite first moment.
Secondly, estimators of the radius and density are defined as the joint minimizer of acontrast function on characteristic functions, and an estimator of the center is proposed
subsequently. These estimators are shown to be consistent. In particular, the radiusestimator is proven to have an almost parametric rate of convergence when the noise
distribution has a finite second moment.
Finally, for d = 2 (i.e., in the case of circular signals), the authors establish that thecenter can be estimated at a nearly parametric rate, and that the signal density on thecircle can be estimated at the same minimax rate as derived in [A. Goldenshluger, J.
Multivariate Anal. 81 (2002), no. 2, 360–375; MR1906385], where it was assumed thatthe noise distribution is known. As in the papers cited above, it is confirmed that theconvergence rate of the signal density does not depend on the tail decay rate of theFourier transform of the error distribution, unlike standard deconvolution problems onRd .
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Results from MathSciNet: Mathematical Reviews on the Webc Copyright American Mathematical Society 2023
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