Asymptotic distribution of the largest off-diagonal entry of correlation matrices
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- by Wang Zhou PDF
- Trans. Amer. Math. Soc. 359 (2007), 5345-5363 Request permission
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 $L_n=\max _{1\le i< j\le p}|\rho _{ij}|$, 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 $l_n=\max _{1\le i< j\le p}|r_{ij}|$, where $r_{ij}$ is Spearman’s rank correlation coefficient between the $i$-th coordinate and the $j$-th coordinate of the population, is derived.References
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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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2327033