Computes the t* statistic corresponding to the tau* population
coefficient introduced by Bergsma and Dassios (2014)
This package allows you to efficiently compute, and perform tests of independence with, the U/V-statistic corresponding to the tau* coefficient described in the paper:
Bergsma, Wicher; Dassios, Angelos. A consistent test of independence based 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 only if the bivariate distribution it is computed upon is independent (under some weak conditions on the bivariate distribution) and is positive otherwise. Since t*, the U-statistic corresponding to tau*, is an unbiased estimator of tau* this gives a consistent test of independence. Computing t* naively results an algorithm that takes O(n^4) time where n is the sample size. Luckily it is possible to compute t* much faster (in O(n^2) time) using the algorithm described in:
Heller, Yair and Heller, Ruth. "Computing the Bergsma Dassios sign-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 Computation of the Bergsma-Dassios Sign Covariance." arXiv preprint arXiv:1504.00964 (2015).
This fast algorithm is implemented in this package. Moreover, the package also uses the results of Nandy, Weihs, and Drton (2016) to allow the use of t* in performing tests of independence. In particular, we provide the function tauStarTest which automates tests of independence using the asymptotic null distribution of t*.
A simple example of computing t* on a independent bivariate normal distribution follows:
> x = rnorm(1000) > y = rnorm(1000) > tStar(x, y)  0.0003637266
Similarly, we may obtain the asymptotic p-value corresponding to a test of independence as follows:
> set.seed(2341) > x = rnorm(1000) > y = rnorm(1000) > tauStarTest(x, y)$pVal  0.5692797
The main functionality of this package is currently included in the functions
tStar (which computes the t* statistic on two input vectors) and
(which performs tests of independence using t*). One may also be interested in
which compute distribution functions, densities, random samples, and quantiles for the asymptotic distribution of t* in different cases.