Cramér-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information
نویسندگان
چکیده
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 CramérRao inequality is a direct consequence of these two inequalities. In this paper the inequalities above are extended to Renyi entropy, p-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 Cramér–Rao inequality is derived as a consequence of these moment and Fisher information inequalities.
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