## The Annals of Statistics

### Necessary and Sufficient Conditions for Inequalities of Cramer-Rao Type

Colin R. Blyth

#### 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

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