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.

- The tStar function now computes the statistic more efficiently using the algorithm described by Heller and Heller (2016). The old algorithm can still be accessed using optional parameters to the function.

- C++ code has been updated so it doesn't cause errors on certain compilers.
- Printing of the output of the tauStarTest function has been fixed.