## The Annals of Statistics

- Ann. Statist.
- Volume 4, Number 5 (1976), 998-1002.

### On the Attainment of the Cramer-Rao Lower Bound

#### Abstract

It is often stated that the variance of an unbiased estimator of a function of a real parameter can attain the Cramer-Rao lower bound only if the family of distributions is a one-parameter exponential family. A rigorous proof of this statement, subject to certain regularity conditions, has been given by Wijsman. However, in general, the statement is not true. Assuming a revised set of regularity conditions it is shown here that there exists a more general class of distributions for which the Cramer-Rao lower bound for the variance is attained for almost all or even all values of the parameter in an interval. The class reduces to the exponential class only by imposing a restriction requiring the absolute continuity in the parameter of a function involving the logarithm of the probability density.

#### Article information

**Source**

Ann. Statist., Volume 4, Number 5 (1976), 998-1002.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343599

**Digital Object Identifier**

doi:10.1214/aos/1176343599

**Mathematical Reviews number (MathSciNet)**

MR413339

**Zentralblatt MATH identifier**

0346.62029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F10: Point estimation

**Keywords**

Cramer-Rao lower bound variance of unbiased estimate attainment of lower bound one-parameter exponential family

#### Citation

Joshi, V. M. On the Attainment of the Cramer-Rao Lower Bound. Ann. Statist. 4 (1976), no. 5, 998--1002. doi:10.1214/aos/1176343599. https://projecteuclid.org/euclid.aos/1176343599