Scale—Invariant Correlation Measures for Discontinuous Distributions

In a recently published paper 1 we were concerned with investigating those properties of two-dimensional distributions that do not depend on the scale by which the random variables are measured. In particular, we pointed out that quantities for measuring the degree of correlation or stochastic dependence of two (random) variables should be so constructed as to be insensitive to changes of the scale of these variables, i.e. scale-invariant. In that paper, a change of scale of the original variables ξ, η meant that the transformations L, g were reversibly unique and left the ordering of the transformed values unchanged.