Entropies of Mixing (EOM) and the Lorenz Order

Abstract

Polynomial nonadditive, or pseudo-additive (PAE), entropies are related to the Shannon entropy in that both are derived from two classes of parent distributions of extreme-value theory, the Pareto and power distributions. The third class is the exponential distribution, corresponding to the Shannon entropy, to which the other two tend as their shape parameters increase without limit. These entropies all belong to a single class of entropies referred to as EOM. EOM is defined as the normalized difference between the dual of the Lorentz function and the Lorenz function. Sufficient conditions for majorization involve finding a separable, Schur-concave function, like the EOM, which increases as the distribution becomes more uniform or less spread out. Lorenz ordering has been associated to the degree in which the Lorenz curve is bent. This criterion is valid for tail distributions, and fails in the case where the distribution is limited on the right. EOM provide criteria for inequality in the Lorenz ordering sense: In the Pareto case, an increase in the shape parameter implies a decrease in inequality and the EOM decreases, whereas for the power distribution an increase in the shape parameter corresponds to an increase in inequality leading to an increase in the EOM. An analogy is drawn between Gauss' invariant distribution for the probability of the fractional part of a continued fraction and the area criterion in Lorenz ordering, analogous to the Gini index criterion. The tendency to approach the invariant distribution, as the number of partial quotients increases without limit, is shown to be analogous to the tendency to approach the invariant area, as the shape parameters increase without limit.