The conventional confidence intervals, also called crisp confidence intervals, using a term from fuzzy set theory, can perform poorly for discrete data. A recent article by Brown, Cai and DasGupta (Ref. 1) reviewed crisp confidence intervals for binomial models. For the binomial distribution and many other discrete distributions, there exist uniformly most powerful (UMP) one-tailed tests and UMP unbiased (UMPU) two-tailed tests, and these tests are optimal procedures. Tests and confidence intervals are dual notions. Hence, randomized confidence intervals based on these tests can achieve their nominal coverage probability and inherit the optimality of these tests. For the binomial distribution, Blyth and Hutchinson (Ref. 2) gave tables for constructing such randomized intervals (for sample sizes up to 50 and coverage probabilities 0.95 and 0.99). Due to the discreteness of the tables, the randomized intervals they produce are not close to exact, hence a computer should now be used instead of these tables. These randomized tests and intervals have been little used in practice, however, because users object to a procedure that can give different answers for the exact same data due to the randomization. It is annoying that two statisticians analyzing exactly the same data and using exactly the same procedure can nevertheless report different results. We can avoid the arbitrariness of randomization while keeping the beautiful theory of these procedures by a simple change of viewpoint to what we call fuzzy and abstract randomized concepts.
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