  The statistical decision theory pioneered by Wald (1950) has used
state-dependent mean loss (risk) to measure the performance of statistical
decision functions across potential samples. We think it evident that
evaluation of performance should respect stochastic dominance, but we do not
see a compelling reason to focus exclusively on mean loss. We think it
instructive to also measure performance by other functionals that respect
stochastic dominance, such as quantiles of the distribution of loss. This paper
develops general principles and illustrative applications for statistical
decision theory respecting stochastic dominance. We modify the Wald definition
of admissibility to an analogous concept of stochastic dominance (SD)
admissibility, which uses stochastic dominance rather than mean sampling
performance to compare alternative decision rules. We study SD admissibility in
two relatively simple classes of decision problems that arise in treatment
choice. We reevaluate the relationship between the MLE, James-Stein, and
James-Stein positive part estimators from the perspective of SD admissibility.
We consider alternative criteria for choice among SD-admissible rules. We
juxtapose traditional criteria based on risk, regret, or Bayes risk with
analogous ones based on quantiles of state-dependent sampling distributions or
the Bayes distribution of loss.
