Quotient correlation: A sample based alternative to Pearson's correlation

TL;DR

This paper introduces quotient correlation as a flexible, sample-based alternative to Pearson's correlation, especially effective in analyzing nonlinear dependence and tail behavior, with new gamma-distributed test statistics and applications in internet traffic data.

Contribution

It develops quotient correlation and related tail dependence measures, providing novel gamma-distributed test statistics and extending correlation analysis to tail behavior and extreme value contexts.

Findings

The quotient correlation effectively captures nonlinear dependence.

The gamma test statistic has high power in tail dependence testing.

Application to internet traffic data demonstrates practical utility.

Abstract

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 $\chi^2$, 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.