Invariant confidence sets with smallest expected measure
A confidence set C(X), \(X\in {\mathcal X}\), is wanted for a function \(\gamma =\gamma(\theta)\in \Gamma\) of a parameter \(\theta\in \Theta\) that subject to covering the true value of \(\gamma\) with probability at least 1- \(\alpha\) (for every \(\theta\in \Theta)\) in some sense minimizes the probabilities of covering wrong values of \(\gamma\). This is made precise by adopting a measure \(m(\cdot,\theta)\) on \(\Gamma\), depending on \(\theta\), and defining the expected measure as \(\int P_{\theta}(\gamma \in C(X)) m(d\gamma,\theta)\). (Actually, this expression is modified slightly to allow for randomization.) It is assumed that there is an invariance group that acts on \({\mathcal X}\) and on \(\Gamma\), and acts transitively on \(\Theta\). If m is chosen to be equivariant and if the confidence set is restricted to be invariant, then the confidence set minimizing the expected measure (subject to the coverage requirement) is independent of \(\theta\). If an invariantly sufficient statistic on \({\mathcal X}\times \Gamma\) exists, then the solution can be expressed explicitly in terms of a Neyman-Pearson-type rule. For measures m that are invariant the minimizing invariant confidence set is minimax among all confidence sets provided conditions similar to those in the Hunt-Stein theorem are satisfied. Several examples are presented, drawn from multivariate analysis, location and scale parameter families, and nonparametrics. It is pointed out that for the purpose of generating simultaneous confidence sets of families of parametric functions of \(\gamma\) several of the confidence sets for \(\gamma\) in the examples have the drawback of not being exact in the reviewer's sense [Multivariate Analysis V, Proc. 5th int. Symp., Pittsburgh 1978, 483-498 (1980; Zbl 0431.62031)]. A few minor corrections to the paper under review have appeared in Ann. Stat. 12, 784 (1984).
scientific article; zbMATH DE number 3852210