The Annals of Statistics

On the Attainment of the Cramer-Rao Lower Bound

V. M. Joshi

Full-text: Open access

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


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