## Electronic Journal of Statistics

### Bounding the maximum of dependent random variables

J. A. Hartigan

#### Abstract

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

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

Dates
First available in Project Euclid: 15 January 2015

https://projecteuclid.org/euclid.ejs/1421330632

Digital Object Identifier
doi:10.1214/14-EJS974

Mathematical Reviews number (MathSciNet)
MR3301303

Zentralblatt MATH identifier
1310.60016

Subjects
Primary: 60E15: Inequalities; stochastic orderings

#### Citation

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

#### References

• [1] Abramowitz, M. and Stegun, I. A., eds. (1972)., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.
• [2] Berman, S. I. (1964). Limit theorems for the maximum term in stationary sequences., Ann. Math. Statist. 35 502–516.
• [3] Bickel, P. J. and Chernoff, H. (1993). Asymptotic distribution of the likelihood ratio statistic in a prototypical non regular problem. In, Statistics and Probability: A Raghu Raj Bahadur Festschrift (J. K. Ghosh, S. K. Mitra, K. R. Parthasarathy and B. Prakasa Rao, eds.) 83–96. Wiley, New York.
• [4] Chernoff, H. and Lander, E. (1995). Asymptotic distribution of the likelihood ratio test that a mixture of two binomials is a single binomial., J. Statist. Plann. Inference 43 19–40.
• [5] Darling, D. and Erdos, P. (1956). A limit theorem for the maximum of normalized sums of independent random variables., Duke Math. J. 23 143–155.
• [6] Gombay, E. and Horvath, L. (1994). An application of the maximum likelihood test to the change-point problem., Stochastic Processes and Their Applications 50 161–171.
• [7] Ghosh, J. K. and Sen, P. K. (1985). On the asymptotic performance of the log likelihood ratio statistics for the mixture model and related results. In, Proceedings of the Berkeley Conference in Honor of J. Neyman and J. Kiefer. 2 789–806.
• [8] Gumbel, E. J. (1941). The return period of flood flows., Ann. Math. Statist. 12 163–190.
• [9] Hartigan, J. A. (1972). A failure of likelihood asymptotics for normal mixtures. In, Proceedings of the Berkeley Conference in Honor of J. Neyman and J. Kiefer. 2 807–810.
• [10] Kim, H. J. and Siegmund, D. (1989). The likelihood ratio test for a change-point in simple linear regression., Biometrika 76 409–423.
• [11] Komlos, J., Major, P. and Tusnady, G. (1976). An approximation of partial sums of independent RV’s and the sample DF. I., Wahrsch verw. Gebiete/Probability Theory and Related Fields 32 111–131.
• [12] Varah, J. M. (1975). A lower bound for the smallest singular value of a matrix., Linear Algebra Appl. 11 3–5.
• [13] Yao, Y.-C. and Davis, R.A. (1986). The asymptotic behavior of the likelihood ratio statistic for testing a shift in mean in a sequence of independent normal variates., Sankhya Ser. A 48 339–353.