Ball divergence


Ball Divergence is a nonparametric two‐sample statistic that quantifies the discrepancy between two probability measures and on a metric space It is defined by integrating the squared difference of the measures over all closed balls in. Let be the closed ball of radius centered at Equivalently, one may set and write The Ball divergence is then defined by
This measure can be seen as an integral of the Harald Cramér's distance over all possible pairs of points. By summing squared differences of and over balls of all scales, BD captures both global and local discrepancies between distributions, yielding a robust, scale-sensitive comparison. Moreover, since BD is defined as the integral of a squared measure difference, it is always non-negative, and if and only if.

Testing for equal distributions

Next, we will try to give a sample version of Ball Divergence. For convenience, we can decompose the Ball Divergence into two parts:
and
Thus
Let denote whether point locates in the ball. Given two independent samples form and form
where means the proportion of samples from the probability measure located in the ball and means the proportion of samples from the probability measure located in the ball. Meanwhile, and means the proportion of samples from the probability measure and located in the ball. The sample versions of and are as follows
Finally, we can give the sample ball divergence
It can be proved that is a consistent estimator of BD. Moreover, if for some, then under the null hypothesis converges in distribution to a mixture of chi-squared distributions, whereas under the alternative hypothesis it converges to a normal distribution.

Properties

  1. The square root of Ball Divergence is a symmetric divergence but not a metric, because it does not satisfy the triangle inequality.
  2. It can be shown that Ball divergence, energy distance test, and MMD are unified within the variogram framework; for details see Remark 2.4 in.

    Homogeneity Test

Ball divergence admits a straightforward extension to the K-sample setting. Suppose are probability measures on a Banach space. Define the K-sample BD by
It then follows from Theorems 1 and 2 that if and only if
By employing closed balls to define a metric distribution function, one obtains an alternative homogeneity measure.
Given a probability measure on a metric space, its metric distribution function is defined by
where is the closed ball of radius centered at, and
If are i.i.d. draws from, the empirical version is
Based on these, the homogeneity measure based on MDF, also called metric Cramér-von Mises is
where be their mixture with weights, and.
The overall MCVM is then
The empirical MCVM is given by
where be an i.i.d. sample from, and
A practical choice for is the median of the squared distances