  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.
