# Test of Independence Using the Tau* Measure

### Description

Performs a (consistent) test of independence between two input vectors using the asymptotic (or permutation based) distribution of the test statistic t*. The asymptotic results hold in the case that x is generated from either a discrete or continous distribution and similarly for y (in particular it is allowed for one to be continuous while the other is discrete). The asymptotic distributions were computed in Nandy, Weihs, and Drton (2016) <http://arxiv.org/abs/1602.04387>.

### Usage

1 | ```
tauStarTest(x, y, mode = "auto", resamples = 1000)
``` |

### Arguments

`x` |
a vector of sampled values. |

`y` |
a vector of sampled values corresponding to x, y must be the same length as x. |

`mode` |
should be one of five possible values: "auto", "continuous", "discrete", "mixed", or "permutation". If "auto" is selected then the function will attempt to automatically determine whether x,y are discrete or continuous and then perform the appropriate asymptotic test. In cases "continuous", "discrete", and "mixed" we perform the associated asymptotic test making the given assumption. Finally if "permutation" is selected then the function runs a Monte-Carlo permutation test for some given number of resamplings. |

`resamples` |
the number of resamplings to do if mode = "permutation". Otherwise this value is ignored. |

### Value

a list with class "tstest" recording the outcome of the test.

### References

Preetam Nandy, Luca Weihs, and Mathias Drton. Large-Sample Theory for the Bergsma-Dassios Sign Covariance. arXiv preprint arXiv:1602.04387. 2016.

### Examples

1 2 3 4 5 6 7 8 9 | ```
set.seed(123)
x = rnorm(100)
y = rnorm(100)
testResults = tauStarTest(x,y)
print(testResults$pVal) # big p-value
y = y + x # make x and y correlated
testResults = tauStarTest(x,y)
print(testResults$pVal) # small p-value
``` |

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