Let \(\prec\) denote the relation of majorization of two discrete distributions. The distribution \(p\) is called more diverse than \(p'\) if \(p\prec p'\). In this sense the uniform distribution is most diverse and a degenerate distribution is least diverse. A numerical function which is monotone increasing w.r.t. the partial order \(\prec\) is called Schur- concave. Both the Gini-Simpson index and Shannon’s entropy have this property and may be considered as a suitable choice for an index of diversity.
The aim of the paper is to introduce a functional which is directly based on the partial order \(\prec\). Starting with the fact that \(p\prec p'\) iff \(p\) is a randomization of \(p'\) by a doubly stochastic matrix \(Q\) the author introduced \[ \rho(p,p')=\inf_ Q\| p-Qp'\|\text{ and } \tilde\rho(p,p')=\rho(p,p')+\rho(p',p). \] The main results of the paper concern isotonic and convexity properties of \(\tilde\rho\). The authors ask for solutions \(H\) of the equation \[ \tilde\rho(p,p')=\lim_{\lambda\to 0}\lambda^{-1}[H(\lambda p+(1- \lambda)p')-\lambda H(p)-(1-\lambda)H(p')] \] to introduce a new measure \(H\) of diversity with the help of the measure of dissimilarity \(\tilde\rho\). A series representation of \(H\) is given. As an application of \(\tilde\rho\), the problem of selecting the most diverse population is considered, where “most diverse” is defined in terms of \(\rho(p_ i,p_ j)\). The selection rule proposed is based on \(\rho(\hat p_ i,\hat p_ j)\). Numerical results for the probability of correct selection are given and compared with those corresponding to procedures based on other diversity indices.