Characterizing optimality among three-decision procedures for directional conclusions

Jones and Tukey (2000) proposed three-decision procedures for directional conclusions in statistical inference, considered as an alternative to the conventional usage of one- and two-tailed significance testing. Moreover, implicit in their suggestions was to consider a procedure to be optimal in case indefinite results were minimized among all procedures with a given control of error. First, we argue by example that this characterization of optimality is not very fruitful when formalized into the strong sense of uniform minimization. Next, imposing a further regularity condition on the comparative class of procedures, we relate the suggested characterization to optimality criteria from test theory (UMP unbiasedness). Similarly, we also consider characterizing optimality in terms of maximizing correct decision rates and minimizing incorrect decision rates. Finally, we demonstrate the applicability of the three considered characterizations with respect to exponential families of distributions.