Hoeffding's independence test

In statistics, Hoeffding's test of independence, named after Wassily Hoeffding, is a test based on the population measure of deviation from independence

${\displaystyle H=\int (F_{12}-F_{1}F_{2})^{2}\,dF_{12}}$

where ${\displaystyle F_{12}}$ is the joint distribution function of two random variables, and ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ are their marginal distribution functions. Hoeffding derived an unbiased estimator of ${\displaystyle H}$ that can be used to test for independence, and is consistent for any continuous alternative. The test should only be applied to data drawn from a continuous distribution, since ${\displaystyle H}$ has a defect for discontinuous ${\displaystyle F_{12}}$, namely that it is not necessarily zero when ${\displaystyle F_{12}=F_{1}F_{2}}$. This drawback can be overcome by taking an integration with respect to ${\displaystyle dF_{1}F_{2}}$. This modified measure is known as Blum–Kiefer–Rosenblatt coefficient.[1]

A paper published in 2008[2] describes both the calculation of a sample based version of this measure for use as a test statistic, and calculation of the null distribution of this test statistic.

See also

• Correlation
• Kendall's tau
• Spearman's rank correlation coefficient
• Distance correlation

References

1. Blum, J.R.; Kiefer, J.; Rosenblatt, M. (1961). "Distribution free tests of independence based on the sample distribution function" (PDF). The Annals of Mathematical Statistics. 32 (2): 485–498. doi:10.1214/aoms/1177705055. JSTOR 2237758.
2. Wilding, G.E., Mudholkar, G.S. (2008) "Empirical approximations for Hoeffding's test of bivariate independence using two Weibull extensions", Statistical Methodology, 5 (2), 160-–170 doi:10.1016/j.stamet.2007.07.002

Primary sources

• Wassily Hoeffding, A non-parametric test of independence, Annals of Mathematical Statistics 19: 293–325, 1948. (JSTOR)
• Hollander and Wolfe, Non-parametric statistical methods (Section 8.7), 1999. Wiley.