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April 2008 Quotient correlation: A sample based alternative to Pearson’s correlation
Zhengjun Zhang
Ann. Statist. 36(2): 1007-1030 (April 2008). DOI: 10.1214/009053607000000866


The quotient correlation is defined here as an alternative to Pearson’s correlation that is more intuitive and flexible in cases where the tail behavior of data is important. It measures nonlinear dependence where the regular correlation coefficient is generally not applicable. One of its most useful features is a test statistic that has high power when testing nonlinear dependence in cases where the Fisher’s Z-transformation test may fail to reach a right conclusion. Unlike most asymptotic test statistics, which are either normal or χ2, this test statistic has a limiting gamma distribution (henceforth, the gamma test statistic). More than the common usages of correlation, the quotient correlation can easily and intuitively be adjusted to values at tails. This adjustment generates two new concepts—the tail quotient correlation and the tail independence test statistics, which are also gamma statistics. Due to the fact that there is no analogue of the correlation coefficient in extreme value theory, and there does not exist an efficient tail independence test statistic, these two new concepts may open up a new field of study. In addition, an alternative to Spearman’s rank correlation, a rank based quotient correlation, is also defined. The advantages of using these new concepts are illustrated with simulated data and a real data analysis of internet traffic.


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Zhengjun Zhang. "Quotient correlation: A sample based alternative to Pearson’s correlation." Ann. Statist. 36 (2) 1007 - 1030, April 2008.


Published: April 2008
First available in Project Euclid: 13 March 2008

zbMATH: 1133.62041
MathSciNet: MR2396823
Digital Object Identifier: 10.1214/009053607000000866

Primary: 60G70 , 62G10 , 62G20 , 62G32 , 62H15 , 62H20

Keywords: necessary condition of tail independence , Nonlinear dependence , tail dependence , testing (tail) independence

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 2 • April 2008
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