The decision theoretic approach to set estimation can be quite useful (see Exercise 9.56) but it can also give some unsettling results, showing the need for thoughtful implementation. Consider again the case of X ~ n(μ, σ2), σ2 unknown, and suppose that we have an interval estimator for μ by C(x) = [x - cs, x + cs], where s2 is an estimator of σ2 independent of X, vS2/σ2 ~ X2v. This is, of course, the usual t interval, one of the great statistical procedures that has stood the test of time. Consider the loss
L((μ, σ), C) = b Length(C) - IC(μ),
similar to that used in Exercise 9.54, but without scaling the length. Construct another procedure C' as

where K is a positive constant. C' does exactly the wrong thing. When s2 is big and there is a lot of uncertainty, we would want the interval to be wide. But C' is empty! Show that we can find a value of K so that
R((μ, σ),C') ‰¤ R(μ, σ),C) for every (μ, σ)
with strict inequality for some (μ, σ).

if s2 K,