Improved confidence estimators for Fieller's confidence sets

Fieller's confidence set C_F for ratios of location parameters, although of great importance in practice, is often cited as an example to criticize frequentist theory. The reason is that the set can consist of the whole parameter space and yet the confidence is γ = 1 – α in any case. In this paper, we study the problem of constructing data-dependent estimators better than γ₊, A reasonable estimator appears to be γ₊, which is one if C_F is the whole parameter space and γ otherwise. By using an estimated confidence approach and a squared-error loss, it is shown that γ₊ dominates γ. The risk improvement of γ₊ over γ can be sizable. Also, by numerically comparing γ₊ with a generalized Bayes estimator γ_L, which is shown to be admissible when one or two ratios are concerned, it is shown that γ₊ is nearly admissible. We also conclude that the common practice of reporting 1 – α only when C_F is not the whole parameter space is nearly admissible.