These functions compute unbiased estimators of squared distance covariance, distance variance, and a bias-corrected estimator of (squared) distance correlation.

1 2 3 | ```
bcdcor(x, y)
dcovU(x, y)
dcovU_stats(Dx, Dy)
``` |

`x` |
data or dist object of first sample |

`y` |
data or dist object of second sample |

`Dx` |
distance matrix of first sample |

`Dy` |
distance matrix of second sample |

The unbiased (squared) dcov is inner product definition of dCov, in the Hilbert space of U-centered distance matrices.

The sample sizes (number of rows) of the two samples must
agree, and samples must not contain missing values. Arguments
`x`

, `y`

can optionally be `dist`

objects;
otherwise these arguments are treated as data.

`dcovU`

returns the unbiased estimator of squared dcov.
`bcdcor`

returns a bias-corrected estimator of squared dcor.

`dcovU_stats`

returns a vector of the components of bias-corrected
dcor: [dCovU, bcdcor, dVarXU, dVarYU].

Unbiased distance covariance (SR2014) corresponds to the biased
(original) *dCov^2*. Since `dcovU`

is an
unbiased statistic, it is signed and we do not take the square root.
For the original distance covariance test of independence (SRB2007,
SR2009), the distance covariance test statistic is the V-statistic
*n V_n^2* (not dCov).
Similarly, `bcdcor`

is bias-corrected, so we do not take the
square root as with dCor.

Maria L. Rizzo mrizzo @ bgsu.edu and Gabor J. Szekely

Szekely, G.J. and Rizzo, M.L. (2014),
Partial Distance Correlation with Methods for Dissimilarities.
*Annals of Statistics*, Vol. 42 No. 6, 2382-2412.

Szekely, G.J., Rizzo, M.L., and Bakirov, N.K. (2007),
Measuring and Testing Dependence by Correlation of Distances,
*Annals of Statistics*, Vol. 35 No. 6, pp. 2769-2794.

http://dx.doi.org/10.1214/009053607000000505

Szekely, G.J. and Rizzo, M.L. (2009),
Brownian Distance Covariance,
*Annals of Applied Statistics*,
Vol. 3, No. 4, 1236-1265.

http://dx.doi.org/10.1214/09-AOAS312

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