## Bernoulli

• Bernoulli
• Volume 25, Number 4B (2019), 3400-3420.

### Rate of divergence of the nonparametric likelihood ratio test for Gaussian mixtures

#### Abstract

We study a nonparametric likelihood ratio test (NPLRT) for Gaussian mixtures. It is based on the nonparametric maximum likelihood estimator in the context of demixing. The test concerns if a random sample is from the standard normal distribution. We consider mixing distributions of unbounded support for alternative hypothesis. We prove that the divergence rate of the NPLRT under the null is bounded by $\log n$, provided that the support range of the mixing distribution increases no faster than $(\log n/\log 9)^{1/2}$. We prove that the rate of $\sqrt{\log n}$ is a lower bound for the divergence rate if the support range increases no slower than the order of $\sqrt{\log n}$. Implications of the upper bound for the rate of divergence are discussed.

#### Article information

Source
Bernoulli, Volume 25, Number 4B (2019), 3400-3420.

Dates
Revised: November 2018
First available in Project Euclid: 25 September 2019

https://projecteuclid.org/euclid.bj/1569398770

Digital Object Identifier
doi:10.3150/18-BEJ1094

Mathematical Reviews number (MathSciNet)
MR4010959

Zentralblatt MATH identifier
07110142

#### Citation

Jiang, Wenhua; Zhang, Cun-Hui. Rate of divergence of the nonparametric likelihood ratio test for Gaussian mixtures. Bernoulli 25 (2019), no. 4B, 3400--3420. doi:10.3150/18-BEJ1094. https://projecteuclid.org/euclid.bj/1569398770

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