## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 1816-1837.

### A one-sample test for normality with kernel methods

Jérémie Kellner and Alain Celisse

#### Abstract

We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence, our procedure may be applied either to test data for normality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 1816-1837.

**Dates**

Received: April 2016

Revised: March 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326429

**Digital Object Identifier**

doi:10.3150/18-BEJ1037

**Mathematical Reviews number (MathSciNet)**

MR3961232

**Zentralblatt MATH identifier**

07066241

**Keywords**

kernel methods maximum mean discrepancy normality test parametric bootstrap reproducing kernel hilbert space

#### Citation

Kellner, Jérémie; Celisse, Alain. A one-sample test for normality with kernel methods. Bernoulli 25 (2019), no. 3, 1816--1837. doi:10.3150/18-BEJ1037. https://projecteuclid.org/euclid.bj/1560326429

#### Supplemental materials

- Supplement to “A one-sample test for normality with kernel methods”. The supplemental article [24] to this article features appendix sections. In Appendix A, normality tests mentioned throughout this article (such as Henze–Zirkler or Energy distance) are briefly introduced. In Appendix B, the proofs of the theorems presented in this article are detailed. Appendix C shows additional experiments. Finally, Appendix D explicitly shows closed-forms expressions for the Fréchet derivative of $N[\theta]$ for practitioners.Digital Object Identifier: doi:10.3150/18-BEJ1037SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.