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May, 1974 Necessary and Sufficient Conditions for Inequalities of Cramer-Rao Type
Colin R. Blyth
Ann. Statist. 2(3): 464-473 (May, 1974). DOI: 10.1214/aos/1176342707


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.


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Colin R. Blyth. "Necessary and Sufficient Conditions for Inequalities of Cramer-Rao Type." Ann. Statist. 2 (3) 464 - 473, May, 1974.


Published: May, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0283.62032
MathSciNet: MR356333
Digital Object Identifier: 10.1214/aos/1176342707

Primary: 62F10
Secondary: 62B99 , 62C15

Keywords: complete family of distributions , Cramer-Rao inequality , minimal sufficient statistic , minimum variance unbiased estimation , quadratic-loss admissibility

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 3 • May, 1974
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