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Asymptotic distribution of the largest off-diagonal entry of correlation matrices


Author: Wang Zhou
Journal: Trans. Amer. Math. Soc. 359 (2007), 5345-5363
MSC (2000): Primary 60F05, 62G20, 62H10
DOI: https://doi.org/10.1090/S0002-9947-07-04192-X
Published electronically: May 11, 2007
MathSciNet review: 2327033
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Abstract: Suppose that we have $ n$ observations from a $ p$-dimensional population. We are interested in testing that the $ p$ variates of the population are independent under the situation where $ p$ goes to infinity as $ n\to \infty$. A test statistic is chosen to be $ L_n=\max_{1\le i< j\le p}\vert\rho_{ij}\vert$, where $ \rho_{ij}$ is the sample correlation coefficient between the $ i$-th coordinate and the $ j$-th coordinate of the population. Under an independent hypothesis, we prove that the asymptotic distribution of $ L_n$ is an extreme distribution of type $ G_1$, by using the Chen-Stein Poisson approximation method and the moderate deviations for sample correlation coefficients. As a statistically more relevant result, a limit distribution for $ l_n=\max_{1\le i< j\le p}\vert r_{ij}\vert$, where $ r_{ij}$ is Spearman's rank correlation coefficient between the $ i$-th coordinate and the $ j$-th coordinate of the population, is derived.


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Additional Information

Wang Zhou
Affiliation: Department of Statistics and Applied Probability, National University of Singapore, Singapore, 117546
Email: stazw@nus.edu.sg

DOI: https://doi.org/10.1090/S0002-9947-07-04192-X
Keywords: Sample correlation matrices, Spearman's rank correlation matrices, Chen-Stein method, moderate deviations.
Received by editor(s): April 25, 2005
Received by editor(s) in revised form: September 5, 2005
Published electronically: May 11, 2007
Additional Notes: The author was supported in part by grants R-155-000-035-112 and R-155-050-055-133/101 at the National University of Singapore
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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