Efficient Computation and Testing of the Bergsma-Dassios Sign Covariance.

TauStar Package

Purpose

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*.

Example

A simple example of computing t* on a independent bivariate normal distribution follows:

> set.seed(2342)
> x = rnorm(1000)
> y = rnorm(1000)
> tStar(x, y)
[1] 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
[1] 0.5692797

Where to go

The main functionality of this package is currently included in the functions tStar (which computes the t* statistic on two input vectors) and tauStarTest (which performs tests of independence using t*). One may also be interested in the functions

1. pHoeffInd, dHoeffInd, rHoeffInd, qHoeffInd
2. pDisHoeffInd, dDisHoeffInd, rDisHoeffInd, qDisHoeffInd
3. pMixHoeffInd, dMixHoeffInd, rMixHoeffInd, qMixHoeffInd

which compute distribution functions, densities, random samples, and quantiles for the asymptotic distribution of t* in different cases.