Unbiased dcov and bias-corrected dcor statistics
These functions compute unbiased estimators of squared distance covariance, distance variance, and a bias-corrected estimator of (squared) distance correlation.
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data or dist object of first sample
data or dist object of second sample
distance matrix of first sample
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
y can optionally be
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).
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.
Szekely, G.J. and Rizzo, M.L. (2009),
Brownian Distance Covariance,
Annals of Applied Statistics,
Vol. 3, No. 4, 1236-1265.
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