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.

Mean-deviation analysis, along with the existing theories of coherent risk measures and dual utility, is examined in the context of the theory of choice under uncertainty, which studies rational preference relations for random outcomes based on different sets of axioms such as transitivity, monotonicity, continuity, etc. An axiomatic foundation of the theory of coherent risk measures is obtained as a relaxation of the axioms of the dual utility theory, and a further relaxation of the axioms are shown to lead to the mean-deviation analysis. Paradoxes arising from the sets of axioms corresponding to these theories and their possible resolutions are discussed, and application of the mean-deviation analysis to optimal risk sharing and portfolio selection in the context of rational choice is considered.

The consistency of law-invariant general deviation measures with concave ordering has been used to generalize the Rao–Blackwell theorem and to develop an approach for reducing minimization of law-invariant deviation measures to minimization of the measures on subsets of undominated random variables with respect to concave ordering. This approach has been applied for constructing the Chebyshev and Kolmogorov inequalities with law-invariant deviation measures—in particular with mean absolute deviation, lower semideviation and conditional value-at-risk deviation. Additionally, an advantage of the Kolmogorov inequality with certain deviation measures has been illustrated in estimating the probability of the exchange rate of two currencies to be within specified bounds.

Abstract:The optimization framework for optimal sensor placement for underwater threat detection has been developed. It considers single-period and multiperiod detection models, each of which includes two components: detection algorithm and optimization problem for sensor placement. The detection algorithms for single-period and multiperiod models are based on likelihood ratio and sequential testing, respectively. For the both models, the optimization problems use the principle of superadditive coverage, which is closely related to energy-based and information-based approaches. An algorithm for quasi-regular sensor placement approximating solutions to the optimization problems has been developed based on corresponding continuous relaxations and a criterion for its applicability has been obtained. Numerical experiments have demonstrated that the algorithm consistently outperforms existing optimization techniques for optimal sensor placement.

Cooperative games with players using different law-invariant deviation measures as numerical representations for their attitudes towards risk in investing to a financial market are formulated and studied. As a central result, it is shown that players (investors) form a coalition (cooperative portfolio) that behaves similar to a single player (investor) with a certain deviation measure. An explicit formula for that deviation measure is obtained. An approach to optimal risk sharing among investors is developed, and a "fair" division of the cooperative portfolio expected gain, belonging to the core of a corresponding cooperative game, is suggested.