Maximum Entropy Principle with General Deviation Measures

An approach to the Shannon and Rényi entropy maximization problems with constraints on the mean and law invariant deviation measure for a random variable has been developed. The approach is based on the representation of law invariant deviation measures through corresponding convex compact sets of nonnegative concave functions. A solution to the problem has been shown to have an alpha-concave distribution (log-concave for Shannon entropy), for which in the case of comonotone deviation measures, an explicit formula has been obtained. As an illustration, the problem has been solved for several deviation measures, including mean absolute deviation (MAD), conditional value-at-risk (CVaR) deviation, and mixed CVaR-deviation. Also, it has been shown that the maximum entropy principle establishes a one-to-one correspondence between the class of alpha-concave distributions and the class of comonotone deviation measures. This fact has been used to solve the inverse problem of finding a corresponding comonotone deviation measure for a given alpha-concave distribution.