Minimax detection of a signal in the heteroscedastic Gaussian white noise

Ermakov, Michael Sergeevich

doi:10.1007/s10958-006-0244-1

Cited by 7 publications

(6 citation statements)

References 6 publications

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“…Lepski and Spokoiny [17] obtained minimax rates of testing over Besov bodies B s,p,q (R) in the irregular case (when 0 < p < 2), see also Ingster and Suslina [13]. Ermakov [10] determines a family of asymptotic minimax tests for testing that the signal belongs to a parametric set against nonparametric sets of alternatives in the heteroscedastic Gaussian white noise. In all these references, asymptotic minimax rates of testing are established.…”

Section: Introductionmentioning

confidence: 99%

Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

Laurent¹,

Loubes²,

Marteau³

2012

Electron. J. Statist.

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The aim of this paper is to establish non-asymptotic minimax rates for goodness-of-fit hypotheses testing in an heteroscedastic setting. More precisely, we deal with sequences (Y j ) j∈J of independent Gaussian random variables, having mean (θ j ) j∈J and variance (σ j ) j∈J . The set J will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to l 2 and l ∞ norms, without assumption on (σ j ) j∈J and on several functions spaces. Our point of view is entirely non-asymptotic.

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Section: Introductionmentioning

confidence: 99%

Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

Laurent¹,

Loubes²,

Marteau³

2012

Electron. J. Statist.

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“…nj satisfy some regularity assumptions, test statistics T n (Y n ) are asymptotically minimax (see [10]) for wider sets of alternatives…”

Section: Necessary and Sufficient Conditions Of Uniform Consistencymentioning

confidence: 99%

“…For specially selected sequences ρ n , ρ n → 0 as n → ∞, in papers [8,10,9] (see Theorems 6.3, 4.3, 5.2 as well) we established uniform consistency of sets Υ n (T n , ρ n ) of alternatives for χ 2 −tests with growing number of cells with increasing of sample size, tests generated L 2 -norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients (moreover asymptotic minimaxity of tests on these sets). In this part of paper we establish uniform consistency of sets Υ n (T, ρ n ) of alternatives for Cramer -von Mises test (see Theorem 7.1.…”

Section: Introductionmentioning

confidence: 99%

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On uniform consistency of nonparametric tests I

Ermakov¹

2020

Preprint

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We provide necessary and sufficient conditions of uniform consistency of nonparametric sets of alternatives for widespread nonparametric tests. Nonparametric sets of alternatives can be defined both in terms of distribution function and in terms of density (or signals in the problem of signal detection in Gaussian white noise). In this part of paper such conditions are provided for χ 2 −tests with increasing number of cells, Cramer-von Mises tests, tests generated L2-norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients.

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“…nj satisfy some regularity assumptions, test statistics T n (Y n ) are asymptotically minimax (see Ermakov [12]) for the wider sets of alternatives…”

Section: Quadratic Test Statisticsmentioning

confidence: 99%

On consistency and inconsistency of nonparametric tests

Ermakov¹

2018

Preprint

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For χ 2 −tests with increasing number of cells, Cramer-von Mises tests, tests generated L 2 -norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients, we find necessary and sufficient conditions of consistency and inconsistency for sequences of alternatives having a given rate of convergence to hypothesis in L 2 -norm. We provide transparent interpretations of these conditions allowing to understand the structure of such consistent sequences. We show that, if set of alternatives is bounded closed center-symmetric convex set U with "small" L 2 -ball removed, then compactness of set U is necessary condition for existence of consistent tests.

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Minimax detection of a signal in the heteroscedastic Gaussian white noise

Cited by 7 publications

References 6 publications

Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

On uniform consistency of nonparametric tests I

On consistency and inconsistency of nonparametric tests

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