Exact first moments of the RV coefficient by invariant orthogonal integration

Abstract

The RV coefficient measures the similarity between two multivariate configurations, and its significance testing has attracted various proposals in the last decades. We present a new approach, the invariant orthogonal integration, permitting to obtain the exact first four moments of the RV coefficient under the null hypothesis.Our proposal can be applied to any multivariate setting endowed with Euclidean distances between the observations. It also covers the weighted setting of observations of unequal importance, where the exchangeability assumption, justifying the usual permutation tests, breaks down.The proposed RV moments express as simple functions of the kernel eigenvalues occurring in the weighted multidimensional scaling of the two configurations (spectral effective dimensionality, spectral skewness and spectral excess kurtosis). The expressions for the third and fourth moments seem original, and explain the marked asymmetry and kurtosis of the RV coefficient. They permit to test the significance of the RV coefficient by Cornish–Fisher cumulant expansion, beyond the normal approximation, as illustrated on a small dataset.The first three moments can be obtained by elementary means, but computing the fourth moment requires a more sophisticated apparatus, the Weingarten calculus for orthogonal groups.