Distance Covariance, Independence, and Pairwise Differences

Figures & data

Table 1: Example 1: $\text{Cov}(|X-X^{\prime }|,|Y-Y^{\prime }|)=0$ but X and Y are dependent.

atom	X	Y	probability
a	−1	1	1/4
b	1	1	1/4
c	0	0.6	1/4
d	0	−1	1/4
atom	$|X-X^{\prime }|$	$|Y-Y^{\prime }|$	probability
$(a,a),(b,b),(c,c),(d,d)$	0	0	1/4
$(c,d)$	0	1.6	1/8
$(a,c),(b,c)$	1	0.4	1/4
$(a,d),(b,d)$	1	2	1/4
$(a,b)$	2	0	1/8
atom	$\Delta (X,X^{\prime })$	$\Delta (Y,Y^{\prime })$	probability
$(a,a),(b,b)$	−1.25	−0.40	1/8
$(d,d)$	−0.25	−2.00	1/16
$(c,c)$	−0.25	−0.40	1/16
$(c,d)$	−0.25	0.40	1/8
$(a,c),(b,c)$	0.25	0.00	1/4
$(a,d),(b,d)$	0.25	0.80	1/4
$(a,b)$	0.75	−0.40	1/8

Figure 1: Example with a distribution on 4 points, from Table 1. Top: plot of Y versus X. Bottom left: plot of pairwise distances $|Y-Y^{\prime }|$ of Y versus those of X. Bottom right: doubly centered distances $\Delta (Y,Y^{\prime })$ of Y versus those of X.

Figure 2: Left: dependent variables generated in Example 3, with horizontal regression line illustrating that X and Y are uncorrelated. Right: Plot of the distance correlation of the standard bivariate t-distribution in Example 4, for a range of $\nu$.