Admissibility of Conditional Confidence Procedures

A formal structure for conditional confidence (cc) procedures is investigated. Underlying principles are a (conditional) frequentist interpretation of the cc coefficient, and highly variable. The latter allows the stated measure of conclusiveness to reflect how intuitively clear-cut the outcome of the experiment is. The methodology may thus answer some criticisms of the Neyman-Pearson-Wald approach, but is in the spirit of the latter and includes it. Example: X has one of k densities f_ wrt. A nonempty set of decisions D_ D is "correct" for state. A nonrandomized cc procedure consists of a pair (, Z) where is a nonrandomized decision function and Z is a conditioning rv. The cc coefficient is _ = P_\^{-1 (D_) Z\}. If X = x₀, we make decision (x₀) with "cc _ (x₀) of being correct if is true"; it is unnecessary, but often a practical convenience (as for un-cc intervals), to have _ independent of. Possible notions of "goodness" are discussed; e. g. , (, Z) at least as good as (, Z) if P_\{ (X) D_ and Gamma_ > t\} p_\delta (X) D_ and _ > t\^ t, , and P_\{_ = 0\} P_\Gamma_ = 0\^. It is proved that cc procedure (, Z) is admissible if the non-cc is admissible. For 2-hypothesis problems the converse is true; otherwise, "star-shaped" partitions of the likelihood ratio space are needed. Other loss structures are also treated.