Admissibility of Conditional Confidence Procedures

Abstract

A formal structure for conditional confidence (cc) procedures is investigated. Underlying principles are a (conditional) frequentist interpretation of the cc coefficient $\Gamma$, and highly variable $\Gamma$. 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 $f_\omega \operatorname{wrt} \nu$. A nonempty set of decisions $D_\omega \subset D$ is "correct" for state $\omega$. A nonrandomized cc procedure consists of a pair $(\delta, Z)$ where $\delta$ is a nonrandomized decision function and $Z$ is a conditioning rv. The cc coefficient is $\Gamma_\omega = P_\omega\{\delta^{-1}(D_\omega)\mid Z\}$. If $X = x_0$, we make decision $\delta(x_0)$ with "cc $\Gamma_\omega(x_0)$ of being correct if $\omega$ is true"; it is unnecessary, but often a practical convenience (as for un-cc intervals), to have $\Gamma_\omega$ independent of $\omega$. Possible notions of "goodness" are discussed; e.g., $(\bar{\bar{\delta}}, \bar{\bar{Z}})$ at least as good as $(\bar{\delta}, \bar{Z})$ if $P_\omega\{\bar{\bar{\delta}}(X)\in D_\omega$ and $\bar{\bar{Gamma}}_\omega > t\} \geqq p_\omega\{\bar{delta}(X) \in D_\omega$ and $\bar{\Gamma}_\omega > t\}^\forall t, \omega,$ and $P_\omega\{\bar{\bar{\Gamma}}_\omega = 0\} \geqq P_\omega\{\bar{Gamma}_\omega = 0\}^\forall\omega$. It is proved that cc procedure $(\delta, Z)$ is admissible if the non-cc $\delta$ 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.