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

Comminges, Laëtitia; Dalalyan, Arnak S.

doi:10.1214/13-ejs766

Cited by 23 publications

(26 citation statements)

References 28 publications

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“…Finally, a last type of results that are related to our problem is the case where the null hypothesis can be parametrised, see e.g. [11] where the authors consider shapes that can be parametrised by a quadratic functional. This approach and their results suggest that the smoothness of the shape of C has an impact on the testing rate.…”

Section: Introductionmentioning

confidence: 99%

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Blanchard¹,

Carpentier²,

Gutzeit³

2018

Electron. J. Statist.

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We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a closed convex subset C of R d . We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension d and variance 1 n giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non-smooth choices for C.MSC 2010 subject classifications: Primary 60K35, 60K35; secondary 60K35.

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Section: Introductionmentioning

confidence: 99%

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Blanchard¹,

Carpentier²,

Gutzeit³

2018

Electron. J. Statist.

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“…So far, we have only focused on the behavior of the test under the null without paying attention on what happens under the alternative. The next theorem fills this gap by establishing the consistency of the test defined by the critical region (13).…”

Section: Resultsmentioning

confidence: 99%

“…We have implemented the proposed testing procedures (13) and (20) in Matlab and carried out a certain number of numerical experiments on synthetic data. The aim of these experiments is merely to show that the methodology developed in the present paper is applicable and to give an illustration of how the different characteristics of the testing procedure, such as the significance level and the power, depend on the noise variance σ 2 * and on the shrinkage weights ν.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Curve registration by nonparametric goodness-of-fit testing

Collier

Dalalyan

2015

Journal of Statistical Planning and Inference

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Please cite this article as: Collier, O., Dalalyan, A.S., Curve registration by nonparametric goodness-of-fit testing. J. Statist. Plann. Inference (2015), http://dx. AbstractThe problem of curve registration appears in many different areas of applications ranging from neuroscience to road traffic modeling. In the present work 1 , we propose a nonparametric testing framework in which we develop a generalized likelihood ratio test to perform curve registration. We first prove that, under the null hypothesis, the resulting test statistic is asymptotically distributed as a chi-squared random variable. This result, often referred to as Wilks' phenomenon, provides a natural threshold for the test of a prescribed asymptotic significance level and a natural measure of lack-of-fit in terms of the p-value of the χ 2 -test. We also prove that the proposed test is consistent, i.e., its power is asymptotically equal to 1. Finite sample properties of the proposed methodology are demonstrated by numerical simulations. As an application, a new local descriptor for digital images is introduced and an experimental evaluation of its discriminative power is conducted.

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“…The consistency of traditional nonparametric tests (Kolmogorov, ω 2 , chi-squared, kernel-based, ...) is also well studied (Lehman and Romano [32], Balakrishnan, Nikulin and Voinov [3], Horowitz and Spokoiny [22], Ermakov [18] and references therein). For hypothesis testing in functional spaces the uniform consistency is explored intensively for approaching nonparametric sets of alternatives (Ingster and Suslina [26], Comminges and Dalalyan [9] and references therein). The conditions of distinquishability in these works satisfy the conditions of this paper although they have completely another form.…”

Section: Introductionmentioning

confidence: 99%

On Consistent Hypothesis Testing

Ermakov

2017

J Math Sci

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Abstract. We establish natural links between different types of consistency: consistency, uniform consistency, pointwise consistency and strong consistency. In particular, we show that the existence of pointwise consistent tests implies the existence of discernible (strongly pointwise consistent) tests. Implementing these results we explore both sufficient conditions and necessary conditions for existence of different types of consistent tests for the problems of hypothesis testing on a probability measure of independent sample, on a mean measure of Poisson process, on a solution of linear ill-posed problem in Gaussian noise, on a solution of deconvolution problem and for a signal detection in Gaussian white noise. In the last three cases we show that necessary conditions and sufficient conditions coincide.

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

Cited by 23 publications

References 28 publications

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Curve registration by nonparametric goodness-of-fit testing

On Consistent Hypothesis Testing

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