Kubokawa, Tatsuya

In estimation of ratio of variances in two normal distributions with unknown means, it has been shown in the literature that simple and crude ratio estimators based on two sample variances are dominated by shrinkage estimators using information contained in sample means. Of these, a natural double shrinkage estimator is the ratio of shrinkage estimators of variances, but its improvement over the crude ratio estimator depends on loss functions, namely, the improvement has not been established except the Stein loss function. In this paper, this dominance property is shown for some convex loss functions including the Stein and quadratic loss functions in the general framework of distributions with positive parameters and shrinkage estimators. The resulting new finding is that the generalized Bayes estimator of the ratio of variances dominates the crude ratio estimator relative to the quadratic loss. The paper also shows that the dominance property of the double shrinkage estimator holds for estimation of the difference of variances, but it does not hold for estimation of the product and sum of variances. Finally, it is demonstrated that the double shrinkage estimators for the ratio, product, sum and differences of variances are connected to estimation of linear combinations of the normal positive means, and the dominance and non-dominance results of the double shrinkage estimators coincide with the corresponding dominance results in estimation of linear combinations of means.