An inclusion‐consistent solution to the problem of absurd confidence statements: 1. Consistent exact confidence‐interval estimation

An example is given of a uniformly most accurate unbiased confidence belt which yields absurd confidence statements with 100% occurrence. In several known examples, as well as in the 100%‐occurrence counterexample, an optimal confidence belt provides absurd statements because it is inclusion‐inconsistent with either a null or an all‐inclusive belt or both. It is concluded that confidence‐theory optimality criteria alone are inadequate for practice, and that a consistency criterion is required. An approach based upon inclusion consistency of belts [C(x) C C C(x) , for some x , implies γ ≤ γ for confidence coefficients] is suggested for exact interval estimation in continuous parametric models. Belt inclusion consistency, the existence of a proper‐pivotal vector [a pivotal vector T(X, θ) such that the effective range of T(x,.) is independent of x] , and the existence of a confidence distribution are proven mutually equivalent. This consistent approach being restrictive, it is shown, using Neyman's anomalous 1954 example, how to determine whether any given parametric function can be estimated consistently and exactly or whether a consistent nonexact solution must be attempted.