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Energy statistics are estimators of the energy distance that depend on thedistances between observations. The idea behind energy statistics is toconsider a statistical potential energy that would parallel Newton'sgravitational potential energy. This statistical potential energy is zero ifand only if a certain null hypothesis relating two distributions holds true. InSzekely and Rizzo(2004), a nonparametric test for equality of two multivariatedistributions was given, based on the Euclidean distance between observations.This test was shown to be effective for high dimensional multivariate data, andwas implemented by an appropriate distribution free permutation test. As anextension of Szekely and Rizzo (2013), here we consider the energy distancebetween to independent random objects X and Y on the object space M, thatadmits an embedding into an Euclidean space. In the case of a Kendall shapespace, we can use its VW-embedding into an Euclidean space of matrices anddefine the extrinsic distance between two shapes as their VW associateddistance. The corresponding energy distance between two distributions ofKendall shapes of k-ads will be called VW-energy distance We test ourmethodology on, to compare the distributions of Kendall shape of the contour ofthe midsagittal section of the Corpus Callossum in normal vs ADHD diagnosedindividuals. Here we use the VW distance between the shapes of two children CCmidsections. Using the CC data coming originally from http://fcon1000.projects.nitrc.org/indi/adhd200/ it appears that the two Kendall shapedistributions are not significantly different.

Testing for the Equality of two Distributions on High Dimensional Object Spaces