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

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

where $F_{12}$ is the joint distribution function of two random variables, and $F_{1}$ and $F_{2}$ are their marginal distribution functions. Hoeffding derived an unbiased estimator of $H$ 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 $H$ has a defect for discontinuous $F_{12}$, namely that it is not necessarily zero when $F_{{12}}=F_{1}F_{2}$.

A recent paper^[1] 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

Notes

1. 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

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.