Decision-theoretic
interval estimation requires the use of loss functions that, typically, take
into account the size and the coverage of the sets. We here consider the class of monotone loss functions that,
under quite general conditions, guarantee Bayesian optimality of highest
posterior probability sets. We focus on three specific families of monotone
losses, namely the linear, the exponential and the rational losses whose
difference consists in the way the sizes of the sets are penalized. Within the
standard yet important set-up of a normal model we propose: 1) an
optimality analysis, to compare the solutions yielded by the alternative
classes of losses; 2) a regret analysis, to evaluate the additional loss of standard
non-optimal intervals of fixed credibility. The article uses an application to
a clinical trial as an illustrative example.
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