Uniform Approximation of Minimax Point Estimates

Abstract

bounded risk. When Conditions A and B stated later hold, it is shown that a unique minimax estimate uo(x) exists and if {un(x)} is any uniformly bounded minimax sequence then the risk functions of un(x) converges uniformly to the risk function of uo(x), so that no almost subminimax estimate can exist which, though not a minimax estimate, has for a wide range of values of the parameter 0, a lower value of the risk than that of the minimax estimate. Under some additional conditions, it is shown that an approximation to the minimax estimate uO(x) in the space 5(.. of functions with bounded risk, may be obtained by the minimax estimate UN(X) in the finite dimensional linear space spanned byN basis vectors vi1, * VN of 5(P), so that the maximum risk of UaN(x) converges to that of uo(x). This may help in finding an approximation to a minimax estimate in non-standard problems, where it is difficult to guess a minimax estimate from invariance or other considerations and specially when the problem is a perturbation of a standard problem. 2. Introduction. It has been pointed out by Hodges and Lehmann [3] and Robbins [8], that in certain cases there exist estimates which are not minimax so that their maximum risks may be slightly greater than that of a minimax estimate but their risk functions are considerably less than that of a minimax estimate, for a range of values of the parameter 0. Such estimates have been called E-minimax or subminimax estimates [8], [11]. Since e is an unspecified small quantity above, the situation may be characterised by the existence of a minimax sequence of estimates {un (x)} (which is a minimax solution in the wide sense in Wald's terminology), whose risk functions do not uniformly converge to that of the minimax estimate uo(x). Frank and Kiefer [1], have given examples from the theory of testing of hypotheses, of such almost subminimax solutions. An example is given in the appendix where for any e however small subminimax estimates of this type, exist for a squared error loss function with unbounded range of the parameter. A second difficulty in the choice of a suitable estimate arises when there is a multiplicity of admissible minimax estimates. In the first part of the paper it is shown that Conditions A and B stated later ensures that a unique minimax estimate uo(x) exists and any minimax sequence of estimates { un (x) I converges with respect to a suitable norm topology (Theorem 1) to uo(x) so that the corresponding risk functions also converge uniformly to