A Contribution to Kiefer's Theory of Conditional Confidence Procedures

The "procedures" discussed in this paper are of the following type: The statistician makes a conventional decision (in a multiple decision problem). He also provides a statement of the guaranteed conditional probability that his decision will be correct, given the observed value of some conditioning random variable. Various admissibility criteria to relate such procedures are proposed. For example, we say that one procedure is better (first sense) than a second if the guaranteed conditional confidence statement using the first is stochastically larger than that using the second, for all possible states of nature. Some ramifications of these admissibility criteria are discussed, and some specific admissible procedures are described for problems with two possible states of nature. In particular, the procedure having the finest possible monotone conditioning and having equal conditional confidence under both states of nature is shown to have many desirable admissibility properties.