Sankhya: The Indian Journal of Statistics
2005, Volume 67, Pt. 3, 499--525
Estimation of a Mean of a Normal Distribution with a Bounded Coefficient of Variation
Tatsuya Kubokawa, University of Tokyo, Japan
SUMMARY. The estimation of a mean of a normal distribution with an unknown variance is addressed under the restriction that the coefficient of variation is within a bounded interval. The paper constructs a class of estimators improving upon the best location-scale equivariant estimator of the mean. It is demonstrated that the class includes three typical estimators: the generalized Bayes estimator based on the uniform prior over the restricted region, the generalized Bayes estimator based on the prior putting mass on the boundary, and a truncated estimator. The non-minimaxity of the best location-scale equivariant estimator is shown in the general location-scale family. When another type of restriction is considered, however, we have a different story that the best location-scale equivariant estimator remains minimax.
AMS (1991) subject classification. Primary 62C10; secondary 62C20, 62F10.
Key words and phrases. Bounded mean, coefficient of variation, decision theory, generalized Bayes estimator, location-scale family, minimaxity, restricted parameters.