Open Access
September, 1976 On the Attainment of the Cramer-Rao Lower Bound
V. M. Joshi
Ann. Statist. 4(5): 998-1002 (September, 1976). DOI: 10.1214/aos/1176343599


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.


Download Citation

V. M. Joshi. "On the Attainment of the Cramer-Rao Lower Bound." Ann. Statist. 4 (5) 998 - 1002, September, 1976.


Published: September, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0346.62029
MathSciNet: MR413339
Digital Object Identifier: 10.1214/aos/1176343599

Primary: 62F10

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

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 5 • September, 1976
Back to Top