This package allows you to efficiently compute, and perform tests ofindependence with, the U/V-statistic corresponding to the tau*coefficient described in the paper:
Bergsma, Wicher; Dassios, Angelos. A consistent test of independencebased on a sign covariance related to Kendall’s tau. Bernoulli 20(2014), no. 2, 1006–1028.
The tau* statistic has the special property that it is 0 if and onlyif the bivariate distribution it is computed upon is independent (undersome weak conditions on the bivariate distribution) and is positiveotherwise. Since t, the U-statistic corresponding to tau, is anunbiased estimator of tau* this gives a consistent test of independence.Computing t* naively results an algorithm that takes O(n^4) time where nis the sample size. Luckily it is possible to compute t* much faster (inO(n^2) time) using the algorithm described in:
Heller, Yair and Heller, Ruth. “Computing the Bergsma Dassiossign-covariance.” arXiv preprint arXiv:1605.08732 (2016).
building off of the O(n^2*log(n)) algorithm of:
Weihs, Luca, Mathias Drton, and Dennis Leung. “Efficient Computationof the Bergsma-Dassios Sign Covariance.” arXiv preprint arXiv:1504.00964(2015).
This fast algorithm is implemented in this package. Moreover, thepackage also uses the results of Nandy, Weihs, and Drton (2016) to allowthe use of t* in performing tests of independence. In particular, weprovide the function tauStarTest which automates tests of independenceusing the asymptotic null distribution of t*.
A simple example of computing t* on a independent bivariate normaldistribution follows:
> set.seed(2342)> x = rnorm(1000)> y = rnorm(1000)> tStar(x, y)[1] 0.0003637266Similarly, we may obtain the asymptotic p-value corresponding to atest of independence as follows:
> set.seed(2341)> x = rnorm(1000)> y = rnorm(1000)> tauStarTest(x, y)$pVal[1] 0.5692797The main functionality of this package is currently included in thefunctionstStar (which computes the t* statistic on twoinput vectors) andtauStarTest (which performs tests ofindependence using t*). One may also be interested in the functions
pHoeffInd,dHoeffInd,rHoeffInd,qHoeffIndpDisHoeffInd,dDisHoeffInd,rDisHoeffInd,qDisHoeffIndpMixHoeffInd,dMixHoeffInd,rMixHoeffInd,qMixHoeffIndwhich compute distribution functions, densities, random samples, andquantiles for the asymptotic distribution of t* in different cases.