Some counterexamples to the theory of confidence intervals

Abstract

Some families of distributions are presented for which certain Neyman confidence intervals have very poor conditional properties. In each case there is a 50% Neyman confidence interval I ( x ) for a parameter θ and a subset A of the sample space such that the conditional probability that I ( X ) covers θ given that X belongs to A is less than 0·2 for all θ and the conditional probability that I ( X ) covers θ given that X is not in A is at least 0·8 for all θ . The families of distributions are somewhat alike. More than one example is presented in order to show that the theory of confidence intervals cannot easily step around the difficulties presented to it.