Electronic Journal of Statistics

Bounding the maximum of dependent random variables

J. A. Hartigan

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Let $M_{n}$ be the maximum of $n$ unit Gaussian variables $X_{1},\ldots,X_{n}$ with correlation matrix having minimum eigenvalue $\lambda_{n}$. Then \[M_{n}\ge\lambda_{n}\sqrt{2\log n}+o_{p}(1).\] As an application, the log likelihood ratio statistic testing for the presence of two components in a 1-dimensional exponential family mixture, with one component known, is shown to be at least $\frac{1} {2}\log\log n(1+o_{p}(n))$ under the null hypothesis that there is only one component.

Article information

Electron. J. Statist., Volume 8, Number 2 (2014), 3126-3140.

First available in Project Euclid: 15 January 2015

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings

Maxima of gaussian processes likelihood ratio test exponential family mixtures


Hartigan, J. A. Bounding the maximum of dependent random variables. Electron. J. Statist. 8 (2014), no. 2, 3126--3140. doi:10.1214/14-EJS974. https://projecteuclid.org/euclid.ejs/1421330632

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