CramÉr–Rao and Moment-Entropy Inequalities for Renyi Entropy and Generalized Fisher Information

Abstract

The moment-entropy inequality shows that a continuous random variable with given second moment and maximal Shannon entropy must be Gaussian. Stam's inequality shows that a continuous random variable with given Fisher information and minimal Shannon entropy must also be Gaussian. The Crame/spl acute/r-Rao inequality is a direct consequence of these two inequalities. In this paper, the inequalities above are extended to Renyi entropy, p/sup th/ moment, and generalized Fisher information. Generalized Gaussian random densities are introduced and shown to be the extremal densities for the new inequalities. An extension of the Crame/spl acute/r-Rao inequality is derived as a consequence of these moment and Fisher information inequalities.