The new rank-based concentration index: Further analysis and properties - Tarald O. Kvålseth (PLoS ONE, 2026)
Additional properties and generalizations are explored for a recently introduced concentration index C K . The C K is based on both the distribution of a set of proportions (probabilities) as well as their ranks. The C K is closely related to and proposed as a preferred alternative to the widely used Q that equals the sum of quadratic terms (proportions). Besides the use of C K and Q as measures of market or industry concentration, with the proportions being market shares, C K or its potential transformations can be used as alternative measures in a variety of real measurement situations for which Q has been applied. The extended analysis of C K includes the proof that C K is a convex function, which makes it capable of decomposition analysis. The sensitivity and transfer effect of C K due to changes in the distribution of the proportions is studied. Derivation is given for the so-called numbers equivalent of C K and for its probability interpretation. Generalizations of C K are considered for changing the relative emphasis of the component proportions. Randomly generated distributions exemplify the limited effect on C K from excluding the smallest proportions that are often unavailable in real situations. Numerical comparisons between C K and other concentration indices are presented for a wide variety of firms or industries. A statistical inference procedure is presented for appropriate situations.