On improving on the minimum risk equivariant estimator of a scale parameter under a lower-bound constraint

Abstract

For scale families with densities ( 1 / θ ) f 1 ( x / θ ) on ( 0 , ∞ ) , we study the problem of estimating θ for scale invariant loss L ( θ , d ) = ρ ( d / θ ) , and under a lower-bound constraint of the form θ ⩾ a . We show that for quite general ( f 1 , ρ ) , the generalized Bayes estimator δ a * with respect to the prior 1 / θ on ( a , ∞ ) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In obtaining the dominance results, we make use of Kubokawa's integral expression of risk difference (IERD) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δ a * . We also make use of results from a companion paper, concerning the estimation of a lower-bounded location parameter, by capitalizing on a connection between location parameter and scale parameter estimation problems. Implications are also given, namely for the problems of estimating a power θ r ( r ≠ 0 ), and of estimating θ under an upper bound constraint θ ⩽ b . Finally, we report briefly on a similar dominance phenomenon for an interval constraint of the form a ⩽ θ ⩽ b with 0 < a < b < ∞ .