On Asymptotically Minimax Nonparametric Detection of Signal in Gaussian White Noise
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ON ASYMPTOTICALLY MINIMAX NONPARAMETRIC DETECTION OF SIGNAL IN GAUSSIAN WHITE NOISE M. S. Ermakov∗
UDC 519.2
For the problem of nonparametric detection of signal in Gaussian white noise, strong asymptotically minimax tests are found. The sets of alternatives are balls in the Besov space Bs2∞ with “small” balls in L2 removed. The balls in the Besov space are defined in terms of orthogonal expansions of functions in trigonometrical basis. Similar result is also obtained for nonparametric hypothesis testing on a solution of ill-posed linear inverse problem with Gaussian random noise. Bibliography: 19 titles.
1. Introduction In the problem of nonparametric signal detection in Gaussian white noise, the rate of consistency of nonparametric tests has been studied for wide class of functional spaces and completely different setups (see [2, 9, 10, 14, 15] and references therein). At the same time asymptotically minimax tests with strong asymptotics of type I and type II error probabilities are known only in the following special cases: if a priori information that the signal belongs to either ellipsoid in L2 (see [3, 4]) or bodies in Besov spaces defined in terms of orthogonal expansions with respect to wavelets (see [10]) is provided, or if the signal satisfies the Lipshitz conditions (see [16]). A goal of the present paper is to pay attention to the fact that asymptotically minimax nonparametric tests with strong asymptotics of type I and type II error probabilities can be obtained also for another sets of alternatives. For orthogonal trigonometrical system of functions, these sets are, for some norm, the balls Bs2∞ (P0 ), s > 0, P0 > 0, in the Besov spaces. Using the balls Bs2∞ (P0 ) in the problems of signal detection is rather natural. The balls Bs2∞ (P0 ) provide reasonable information on the smoothness of signal. For the most widespread nonparametric tests, these balls are the largest sets (maxisets) with given rate of consistency [6]. Maxisets are intensively explored in nonparametric estimation (see [12,13,18] and references therein). In particular, Kerkyacharian and Picard [12] have shown that the balls Bs2∞ (P0 ) in a Besov space are maxisets for linear estimators. For the balls Bs2∞ (P0 ), asymptotically minimax estimators [5] are the estimators of the Tikhonov regularizing algorithm. In nonparametric hypothesis testing, the maxisets have been explored in [1, 6]. In the present paper, the statement of the problem is as follows. We observe a realization of a random process Yn (t), t ∈ [0, 1), defined by the stochastic differential equation σ (1.1) d Yn (t) = f (t) dt + √ dw(t), t ∈ [0, 1], σ > 0, n where f ∈ L2 (0, 1) is unknown signal and dw(t) is Gaussian white noise. ∗
St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 124–138. Original article submitted November 7, 2018. 78 1072-3374/20/2511-0078 ©2020 Springer Science+Business Media, LLC
We have a priori information that ∞ 2s 2 1 ¯ s (P0 ) = f : f =
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