Projection correlation between two random vectors

We propose the use of projection correlation to characterize dependence between two random vectors. Projection correlation has several appealing properties. It equals zero if and only if the two random vectors are independent, it is not sensitive to the dimensions of the two random vectors, it is invariant with respect to the group of orthogonal transformations, and its estimation is free of tuning parameters and does not require moment conditions on the random vectors. We show that the sample estimate of the projection correction is $n$-consistent if the two random vectors are independent and root-$n$-consistent otherwise. Monte Carlo simulation studies indicate that the projection correlation has higher power than the distance correlation and the ranks of distances in tests of independence, especially when the dimensions are relatively large or the moment conditions required by the distance correlation are violated.