TY - JOUR
T1 - Maximum entropy principle with general deviation measures
AU - Grechuk, Bogdan
AU - Molyboha, Anton
AU - Zabarankin, Michael
PY - 2009/5
Y1 - 2009/5
N2 - 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.
AB - 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.
KW - Deviation measures
KW - Maximum entropy principle
KW - Rényi entropy
KW - Shannon entropy
UR - http://www.scopus.com/inward/record.url?scp=67649997299&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=67649997299&partnerID=8YFLogxK
U2 - 10.1287/moor.1090.0377
DO - 10.1287/moor.1090.0377
M3 - Article
AN - SCOPUS:67649997299
SN - 0364-765X
VL - 34
SP - 445
EP - 467
JO - Mathematics of Operations Research
JF - Mathematics of Operations Research
IS - 2
ER -