The restricted regularity of admissible minimax criteria for many well known parametric statistical problems

Summary

The stability of test selection criteria is a question that was raised byLeCam in 1964 [seeLeCam, 1964], but which seems to have received very little subsequent attention. Roughly, the problem posed byLeCam was to decide whether a given criterion is regular at a given statistical problem, i.e. given that the criterion prescribes the test δ₀ for the statistical problem, (Λ₀,D ₀,R ₀), whether it prescribes a best δ which is nearly the same as δ₀ for any statistical problem, (Λ, D,R), near to (Λ₀, D₀,R ₀).

A more precise formulation ofLeCam's question is one of the objectives of this paper. This involves defining a suitable topology on the domain of problems where the criterion in question is to be applied. Elementary minimax results of a general nature are proved, and counter examples are given to show that the assumptions used are not completely superfluous. The generality of these results seems to lend support to the authors opinion, that the method of formulation is suitable for treating minimax questions. We then consider the standard elementary problems of parametric statistics, (monotone likelihood ratio, general exponential family) and prove that the admissible minimax criterion is regular with respect to certain types of perturbations in these problems. The treatment is, in a sense, restricted and departs fromLeCam in holding the risk function essentially constant while varying the parameter space. That the problem variations considered may be of some practical interest and that the general formulations used are not completely unrealistic is shown at the conclusion of the paper where we apply a rest, icted regularity theorem to obtain regularity results for the binomial case.