Nonparametric signal detection with small type I and type II error probabilities

Ermakov, Mikhail

doi:10.1007/s11203-010-9048-5

Cited by 7 publications

(6 citation statements)

References 22 publications

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“…Results (10) and (11) have been obtained within a framework of efficient inference for moderate deviation probabilities, cf. Ermakov [10], [12]. Recall that in our setting c n = o(n K ) for every K > 0, so that the strong asymptotics (11) holds in the nonadaptive setting.…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

Sharp adaptive nonparametric testing for Sobolev ellipsoids

Ji,

Nussbaum

2012

Preprint

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We consider testing for presence of a signal in Gaussian white noise with intensity n −1/2 , when the alternatives are given by smoothness ellipsoids with an L 2 -ball of (squared) radius ρ removed. It is known that, for a fixed Sobolev type ellipsoid Σ(β, M ) of smoothness β and size M , a squared radius ρ ≍ n −4β/(4β+1) is the critical separation rate, in the sense that the minimax error of second kind over α-tests stays asymptotically between 0 and 1 strictly (Ingster [22]). In addition, Ermakov [9] found the sharp asymptotics of the minimax error of second kind at the separation rate. For adaptation over both β and M in that context, it is known that a log log-penalty over the separation rate for ρ is necessary for a nonzero asymptotic power. Here, following an example in nonparametric estimation related to the Pinsker constant, we investigate the adaptation problem over the ellipsoid size M only, for fixed smoothness degree β. It is established that the sharp risk asymptotics can be replicated in that adaptive setting, if ρ → 0 more slowly than the separation rate. The penalty for adaptation here turns out to be a sequence tending to infinity arbitrarily slowly.

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Section: Introduction and Main Resultsmentioning

confidence: 96%

Sharp adaptive nonparametric testing for Sobolev ellipsoids

Ji,

Nussbaum

2012

Preprint

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“…The major part of the statistical inference for nonparametric hypotheses testing was developed for the Gaussian white noise model (GWNM) and its equivalent formulation as Gaussian sequence model (GSM). As recent references for the problem of testing a simple hypothesis in these models, we cite (Ermakov, 2011, Ingster et al, 2012, where the reader may find further pointers to previous work. In the present work, the null hypothesis defined by ( 5) is composite and nonparametric.…”

Section: Relation To Previous Workmentioning

confidence: 99%

Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression

Comminges¹,

Dalalyan²

2013

Electron. J. Statist.

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International audienceWe consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional $Q$, the null hypothesis states that the regression function $f$ satisfies the constraint $Q[f]=0$, while the alternative corresponds to the functions for which $Q[f]$ is bounded away from zero. On the one hand, we provide minimax rates of testing and the exact separation constants, along with a sharp-optimal testing procedure, for diagonal and nonnegative quadratic functionals. We consider smoothness classes of ellipsoidal form and check that our conditions are fulfilled in the particular case of ellipsoids corresponding to anisotropic Sobolev classes. In this case, we present a closed form of the minimax rate and the separation constant. On the other hand, minimax rates for quadratic functionals which are neither positive nor negative makes appear two different regimes: ''regular'' and ''irregular''. In the ''regular" case, the minimax rate is equal to $n^{-1/4}$ while in the ''irregular'' case, the rate depends on the smoothness class and is slower than in the ''regular'' case. We apply this to the issue of testing the equality of norms of two functions observed in noisy environments

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“…Further development of distance (semiparametric) method has obtained in Horowitz and Spokoiny [17] and Ermakov [10,11,13]. For χ 2 −tests with increasing number of cells, tests generated L 2 -norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients asymptotic minimaxity of tests has been established (see also Theorems 8.1,8.2,8.3).…”

Section: Introductionmentioning

confidence: 96%

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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Nonparametric signal detection with small type I and type II error probabilities

Abstract: Signal detection, Nonparametric hypothesis testing, Kernel estimator, Large deviations,

Cited by 7 publications

References 22 publications

Sharp adaptive nonparametric testing for Sobolev ellipsoids

Sharp adaptive nonparametric testing for Sobolev ellipsoids

Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression

On consistency and inconsistency of nonparametric tests

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