The problem of Bayes estimation of the scale parameter is considered. Lower bounds for the asymptotic Bayes risk are given as the restricted parameter space increases to the positive half-line. The results are next applied to establish the second-order minimax estimator of the scale parameter. Surprisingly, the least favorable distribution coincides with that for the corresponding location parameter problem.
"Lower Bounds for the Asymptotic Bayes Risk in the Scale Model (with an Application to the Second-Order Minimax Estimation)." Ann. Statist. 22 (4) 1831 - 1839, December, 1994. https://doi.org/10.1214/aos/1176325759