## Electronic Journal of Statistics

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

#### Abstract

We 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 problem of testing the equality of Sobolev norms of two functions observed in noisy environments.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 146-190.

Dates
First available in Project Euclid: 24 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1359041588

Digital Object Identifier
doi:10.1214/13-EJS766

Mathematical Reviews number (MathSciNet)
MR3020417

Zentralblatt MATH identifier
1337.62090

Subjects
Primary: 62G08: Nonparametric regression 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

#### Citation

Comminges, Laëtitia; Dalalyan, Arnak S. Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression. Electron. J. Statist. 7 (2013), 146--190. doi:10.1214/13-EJS766. https://projecteuclid.org/euclid.ejs/1359041588

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