Necessary and Sufficient Conditions for Inequalities of Cramer-Rao Type

Abstract

For a random variable $X$ with possible distributions indexed by a parameter $\theta$, and for real-valued $T = T(X)$ and $V = V(X, \theta)$ with $\operatorname{Var} T < \infty$ and $0 < \operatorname{Var} V < \infty$, Schwarz's inequality gives $\operatorname{Var} T \geqq \{\operatorname{Cov} (T, V)\}^2/\operatorname{Var} V$. Necessary and sufficient conditions are given for this inequality to be of Cramer-Rao type: $\operatorname{Var} T \geqq \{a_m(\theta)\}^2/\operatorname{Var} V$ where $m(\theta)$ is a notation for $ET$ and $a_m(\theta)$ is a notation for $\operatorname{Cov} (T, V)$. Specialized to $V = \{\partial p\theta(X)/\partial\theta\}/p_\theta(X)$, where $p_\theta$ is a probability density function for $X$, these conditions are necessary and sufficient for validity of the Cramer-Rao inequality. The use of these inequalities in proving an estimator minimum variance unbiased is shown to be superfluous. The use of these inequalities in proving admissibility is discussed, with examples.