The Annals of Statistics

Necessary and Sufficient Conditions for Inequalities of Cramer-Rao Type

Colin R. Blyth

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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.

Article information

Source
Ann. Statist., Volume 2, Number 3 (1974), 464-473.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342707

Digital Object Identifier
doi:10.1214/aos/1176342707

Mathematical Reviews number (MathSciNet)
MR356333

Zentralblatt MATH identifier
0283.62032

JSTOR
links.jstor.org

Subjects
Primary: 62F10: Point estimation
Secondary: 62C15: Admissibility 62B99: None of the above, but in this section

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

Citation

Blyth, Colin R. Necessary and Sufficient Conditions for Inequalities of Cramer-Rao Type. Ann. Statist. 2 (1974), no. 3, 464--473. doi:10.1214/aos/1176342707. https://projecteuclid.org/euclid.aos/1176342707


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