ISSN 0278-6419, Moscow University Computational Mathematics and Cybernetics, 2018, Vol. 42, No. 2, pp. 85–88.
Allerton Press, Inc., 2018.
Original Russian Text
O.V. Shestakov, 2018, published in Vestnik Moskovskogo Universiteta, Seriya 15: Vychislitel’naya Matematika i Kibernetika, 2018, No. 2,
Limit Theorems for Risk Estimate
in Models with Non-Gaussian Noise
O. V. Shestakov
Faculty of Computational Mathematics and Cybernetics,
Moscow State University, Moscow, 119991 Russia;
Institute of Informatics Problems, Federal Research Center “Computer Science and Control”,
Russian Academy of Sciences, Moscow, 119333 Russia
Received November 15, 2017
Abstract—The problem of constructing an estimate of a signal function from noisy observations,
assuming that this function is uniformly Lipschitz regular, is considered. The thresholding of
empirical wavelet coeﬃcients is used to reduce the noise. As a rule, it is assumed that the noise
distribution is Gaussian and the optimal parameters of thresholding are known for various classes of
signal functions. In this paper a model of additive noise whose distribution belongs to a fairly wide
class, is considered. The mean-square risk estimate of thresholding is analyzed. It is shown that
under certain conditions, this estimate is strongly consistent and asymptotically normal.
Keywords: thresholding, non-Gaussian noise, mean-square risk estimate.
Many algorithms of signal and image processing are based on possibility of a sparse representation
of the useful signal function in a certain basis. For a fairly wide class of functions, such sparseness is
ensured by using wavelet bases . This enables us to eﬀectively separate noise from the useful signal
and remove it using simple thresholding procedures [2–5]. A classical observation model assumes the
presence of white Gaussian noise. The properties of estimates obtained by thresholding have been well
studied, and we know the order of the mean-square risk of such procedures is found to be close to
optimal . Some results also testify to the asymptotic behavior of a mean-square estimate constructed
from noisy observations of a signal function . The strong consistency and the asymptotic normality of
this estimate were demonstrated in [7, 8].
A wider class of possible noise distributions, especially ones with heavier tails than a Gaussian
distribution, was considered in . The parameters of so-called universal thresholding were calculated
for this class, and it was shown the order of the mean-square risk was close to minimal with an
accuracy up to the logarithm of the number of observations in a certain power, which depends on the
distribution parameters. In this work, we prove the strong inconsistency and asymptotic normality of the
mean-square risk estimate of universal thresholding in a model with a non-Gaussian noise distribution,
assuming the signal function belongs to the Lipschitz class with a certain index.
2. THRESHOLDING IN A DATA MODEL WITH ADDITIVE NOISE
Let signal function f be deﬁned on a certain interval [a, b] and uniformly Lipschitz regular with an
exponent γ>0.Inpractice,f is given in discrete samples. We assume that the number of these samples
for a certain J>0. After the discrete wavelet transform of the signal, set of wavelet coeﬃcients