Fast Rank-Based Independence Testing
independence provides three ranking-based nonparametric tests
for the independence of two continuous variables X and Y:
the classical Hoeffding's D test:
a refined variant of it, named R:
the Bergsma-Dassios T* sign covariance:
The first test is consistent assuming an absolutely continuous joint distribution, i.e., the population coefficient D=0 iff the variables are independent. The latter two are consistent under no restriction on the distribution.
Given an iid sample (X1,Y1),...,(Xn,Yn), all three statistics are computed in time O(n log n) improving upon previous implementations. The statistics R and T* are computed by a new algorithm, following work of Even-Zohar and Leng. It is based on the fast counting of certain patterns in the permutation that relates the ranks of X and Y. See [arxiv:2010.09712] and references therein.
Chaim Even-Zohar <email@example.com>
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library(independence) ## independent set.seed(123) xs = rnorm(10000) ys = rnorm(10000) hoeffding.D.test(xs,ys) hoeffding.refined.test(xs,ys) tau.star.test(xs,ys) ## dependent, even though uncorrelated set.seed(123) xs = rnorm(10000,0,3001:13000) ys = rnorm(10000,0,3001:13000) hoeffding.D.test(xs,ys) hoeffding.refined.test(xs,ys) tau.star.test(xs,ys) ## dependent but not absolutely continuous, fools Hoeffding's D set.seed(123) xs = runif(200) f = function(x,y) ifelse(x>y, pmin(y,x/2), pmax(y,(x+1)/2)) ys = f(xs,runif(200)) hoeffding.D.test(xs,ys) hoeffding.refined.test(xs,ys) tau.star.test(xs,ys)
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