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)
[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
```

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

`pHoeffInd`

,`dHoeffInd`

,`rHoeffInd`

,`qHoeffInd`

`pDisHoeffInd`

,`dDisHoeffInd`

,`rDisHoeffInd`

,`qDisHoeffInd`

`pMixHoeffInd`

,`dMixHoeffInd`

,`rMixHoeffInd`

,`qMixHoeffInd`

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

- Fixed some failing tests caused by a recent change in the way R's sample function is implemented.