PLoS One. 2026 Feb 24;21(2):e0343034. doi: 10.1371/journal.pone.0343034. eCollection 2026.
ABSTRACT
Additional properties and generalizations are explored for a recently introduced concentration index CK. The CK is based on both the distribution of a set of proportions (probabilities) as well as their ranks. The CK 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 CK and Q as measures of market or industry concentration, with the proportions being market shares, CK 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 CK includes the proof that CK is a convex function, which makes it capable of decomposition analysis. The sensitivity and transfer effect of CK due to changes in the distribution of the proportions is studied. Derivation is given for the so-called numbers equivalent of CK and for its probability interpretation. Generalizations of CK are considered for changing the relative emphasis of the component proportions. Randomly generated distributions exemplify the limited effect on CK from excluding the smallest proportions that are often unavailable in real situations. Numerical comparisons between CK and other concentration indices are presented for a wide variety of firms or industries. A statistical inference procedure is presented for appropriate situations.
PMID:41734214 | DOI:10.1371/journal.pone.0343034
