{{Short description|Statistical measure}} In statistics, '''Hoeffding's test of independence''', named after Wassily Hoeffding, is a test based on the population measure of deviation from independence
:<math>H = \int (F_{12}-F_1F_2)^2 \, dF_{12} </math>
where <math>F_{12}</math> is the joint distribution function of two random variables, and <math>F_1</math> and <math>F_2</math> are their marginal distribution functions. Hoeffding derived an unbiased estimator of <math>H</math> 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 <math>H</math> has a defect for discontinuous <math>F_{12}</math>, namely that it is not necessarily zero when <math>F_{12}=F_1F_2</math>. This drawback can be overcome by taking an integration with respect to <math>dF_1F_2</math>. This modified measure is known as Blum–Kiefer–Rosenblatt coefficient.<ref>{{Cite journal|last1=Blum|first1=J.R.|last2=Kiefer|first2=J.|last3=Rosenblatt|first3=M.|title=Distribution free tests of independence based on the sample distribution function|url=https://www.jstor.org/stable/pdf/2237758.pdf|journal=The Annals of Mathematical Statistics|year=1961 |volume=32|pages=485–498|number=2|doi=10.1214/aoms/1177705055 |jstor=2237758 }}</ref>
A paper published in 2008<ref>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}}</ref> 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==
{{Portal|Mathematics}} * Correlation * Kendall's tau * Spearman's rank correlation coefficient *Distance correlation
== References == {{Reflist}}
==Primary sources== * Wassily Hoeffding, A non-parametric test of independence, ''Annals of Mathematical Statistics'' '''19''': 293–325, 1948. ([https://www.jstor.org/stable/2236021 JSTOR]) * Hollander and Wolfe, Non-parametric statistical methods (Section 8.7), 1999. Wiley.
{{DEFAULTSORT:Hoeffding's Independence Test}} Category:Covariance and correlation Category:Nonparametric statistics Category:Statistical tests
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