Lorenz theory of symmetric relative concentration and similarity, incorporating variable array length

Abstract

This paper extends the Lorenz theory, developed in [L. Egghe and R. Rousseau. Symmetric and asymmetric theory of relative concentration and applications. Scientometrics 52(2), 261-290, 2001], so that it can deal with comparing arrays of variable length. We show that in this case we need to divide the Lorenz curves by certain types of increasing functions of the array length N. We then prove that, in this theory, adding zeros to two arrays, increases their similarity, a property that is not satisfied by the Pearson correlation coefficient. Among the many good similarity measures, satisfying the developed Lorenz theory, we deduce the correlation coefficient of Spearman, hence showing that this measure can be used as a good measure of symmetric relative concentration (or similarity).