Multivariate nonparametric divergence (or distance) between distributions

Question

For example, we could say I have two fruit classes (oranges and apples) and for each one I measured different statistics of interest, for example: width, height, sugar, water... of a lot of fruit units. Therefore, for each of the statistics I have a probability distribution based on the, let's say, 100,000 different fruits measured.

           Width    Height  Sugar
Apple       X1       Y1      Z1
Orange      X2       Y2      Z2

In this case, the distributions are probably normal distributed, but in my case they are not. I don't know the type of distribution.

I would like to compare them in order to see «how similar» they are. I searched and I found a lot of tests for checking the divergence (or distance) between distributions (Kullback-Leibler, Kolmogorov-Smirnov, Hellinger, Jensen-Shannon...). The problem is that with them I can only compare two distributions (for example apple and orange width). It would be better to take all the statistics of interest together for the comparison.

Therefore, I would like to know a Multivariate nonparametric technique for comparing the divergences/distances between the distributions that gives a number. For example, DIST(X1,Y1,Z1 | X2,Y2,Z2) = value between [0,1], meaning that there is no divergence or there is a lot, respectively.

Answer 1

Multi-response Permutation Procedures (MRPP) is one possibility, with the two groups being apples and oranges. Note that both the degree of overlap and overall within-group variation matter for the final difference between groups, so it is possible to have two groups that overlap completely be different if they differ in within-group distance. So best to use an ordination technique such as NMDS to visualize the results. Significance value is obtained via permutation. http://www.inside-r.org/packages/cran/vegan/docs/mrpp