Adaptive minimax testing in the discrete regression scheme

Gayraud, Ghislaine; Pouet, Christophe

doi:10.1007/s00440-005-0445-4

Cited by 25 publications

(36 citation statements)

References 20 publications

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“…This asymptotic pivotality property paves the way to non-trivial statistical applications. A generalized Wilks phenomenon has been reported in several settings [16,18,35]. But Fan et al observe:…”

Section: Introductionmentioning

confidence: 94%

A high-dimensional Wilks phenomenon

Boucheron

Massart

2010

Probab. Theory Relat. Fields

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A theorem by Wilks asserts that in smooth parametric density estimation the difference between the maximum likelihood and the likelihood of the sampling distribution converges toward a Chi-square distribution where the number of degrees of freedom coincides with the model dimension. This observation is at the core of some goodness-of-fit testing procedures and of some classical model selection methods. This paper describes a non-asymptotic version of the Wilks phenomenon in bounded contrast optimization procedures. Using concentration inequalities for general functions of independent random variables, it proves that in bounded contrast minimization (as for example in Statistical Learning Theory), the difference between the empirical risk of the minimizer of the true risk in the model and the minimum of the empirical risk (the excess empirical risk) satisfies a Bernstein-like inequality where the variance term reflects the dimension of the model and the scale term reflects the noise conditions. From a mathematical statistics viewpoint, the significance of this result comes from the recent observation that when using model selection via penalization, the excess empirical risk represents a minimum penalty if non-asymptotic guarantees concerning prediction error are to be provided. From the perspective of empirical process theory, this paper describes a concentration inequality for the supremum of a bounded non-centered (actually non-positive) empirical process. Combining the now classical analysis of M-estimation (building on Talagrand's inequality for suprema of 123 406 S. Boucheron, P. Massart empirical processes) and versatile moment inequalities for functions of independent random variables, this paper develops a genuine Bernstein-like inequality that seems beyond the reach of traditional tools.

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Section: Introductionmentioning

confidence: 94%

A high-dimensional Wilks phenomenon

Boucheron

Massart

2010

Probab. Theory Relat. Fields

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“…As for the fixed design regression model, Härdle and Kneip in [9] state a first minimax result over some particular Sobolev classes. Gayraud and Pouet in [7] consider the more general problem of testing composite null hypotheses such as "f belongs to some parametric family". They give the minimax rates of testing for this problem with respect to the L 2 −norm over some Hölder classes and prove that these rates are of the same order as those obtained for simple hypotheses in the Gaussian white noise model (see [12]).…”

Section: ρ(S α δ σ) = Infmentioning

confidence: 99%

Adaptive tests for periodic signal detection with applications to laser vibrometry

Fromont

Lévy‐Leduc

2006

ESAIM: PS

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Abstract.Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l2−distance between the signal and zero so that the detection is possible with prescribed probabilities of error. Then, we propose a testing procedure which is available when the variance of the noise is unknown and which does not use any prior information about the smoothness degree or the period of the signal. We prove that it is adaptive in the sense that it achieves, up to a possible logarithmic factor, the minimax separation rate over various periodic Sobolev balls simultaneously. The originality of our approach as compared to related works on the topic of signal detection is that our testing procedure is sensitive to the periodicity assumption on the signal. A simulation study is performed in order to evaluate the effect of this prior assumption on the power of the test. We do observe the gains that we could expect from the theory. At last, we turn to the application to target detection by laser vibrometry that we had in view.Mathematics Subject Classification. 62G10, 62G08, 62G20.

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“…Some functional estimation problems, whose goal is to infer f (θ) for a given function f , are also related to some composite-composite testing problems. In fact, some work on functional problems [7,8,10,19,32] and adaptive confidence regions (e.g. [9,11,23,37]) have lead to progress in the understanding of such testing problems.…”

Section: Related Literaturementioning

confidence: 99%

“…We will prove this claim by contraposition. In the sequel, we assume that both (19) and (24) for all q ≤ ∆ are not satisfied. The analysis is divided into 5 cases depending on the values of k 0 , ∆ and n.…”

Section: B24 Analysis Of T C αKmentioning

confidence: 99%

Adaptive estimation of the sparsity in the Gaussian vector model

Carpentier¹,

Verzélen²

2019

Ann. Statist.

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Consider the Gaussian vector model with mean value θ. We study the twin problems of estimating the number θ 0 of non-zero components of θ and testing whether θ 0 is smaller than some value. For testing, we establish the minimax separation distances for this model and introduce a minimax adaptive test. Extensions to the case of unknown variance are also discussed. Rewriting the estimation of θ 0 as a multiple testing problem of all hypotheses { θ 0 ≤ q}, we both derive a new way of assessing the optimality of a sparsity estimator and we exhibit such an optimal procedure. This general approach provides a roadmap for estimating the complexity of the signal in various statistical models.

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Adaptive minimax testing in the discrete regression scheme

Cited by 25 publications

References 20 publications

A high-dimensional Wilks phenomenon

A high-dimensional Wilks phenomenon

Adaptive tests for periodic signal detection with applications to laser vibrometry

Adaptive estimation of the sparsity in the Gaussian vector model

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