Cram\'{e}r-Rao-type Bound and Stam's Inequality for Discrete Random Variables

TL;DR

This paper extends classical bounds like the Cramér-Rao and Stam's inequality to discrete random variables by defining Fisher information in the discrete setting, providing new theoretical tools.

Contribution

It introduces a discrete Fisher information and derives corresponding Cramér-Rao-type bound and Stam's inequality for discrete variables.

Findings

Established a discrete Fisher information concept.

Derived a discrete Cramér-Rao-type bound.

Proved a discrete Stam's inequality.

Abstract

The variance and the entropy power of a continuous random variable are bounded from below by the reciprocal of its Fisher information through the Cram\'{e}r-Rao bound and the Stam's inequality respectively. In this note, we introduce the Fisher information for discrete random variables and derive the discrete Cram\'{e}r-Rao-type bound and the discrete Stam's inequality.