A general technique is developed for improving upon inadmissible estimators of natural parameters (or integral powers thereof) from continuous exponential families. The technique is to reduce the problem to the study of a differential inequality. Typical differential inequalities are presented and solved. Explicit results are given for the simultaneous estimation of gamma scale parameters (and their inverses) for a variety of natural loss functions. Surprising behavior is observed for many of the estimators improving upon "standard" estimators. For squared error loss (and any continuous exponential family) it is shown explicitly how to establish inadmissibility of an estimator and construct improved estimators.
"Improving on Inadmissible Estimators in Continuous Exponential Families with Applications to Simultaneous Estimation of Gamma Scale Parameters." Ann. Statist. 8 (3) 545 - 571, May, 1980. https://doi.org/10.1214/aos/1176345008