On the Theory of Ban Estimates

Abstract

The notion of best asymptotically normal estimates--BAN estimates for short--was introduced by Neyman [8] in the multinomial case. Applications have been made in biological problems, notably in bio-assay [2], [4], [5]. Generalizations of Neyman's work have been made by Barankin and Gurland [1], Chiang [3] and Ferguson [5]. The usual theory of BAN estimates requires differentiability of the estimates, and imposes rather strong conditions on certain functions given in advance (the functions $\zeta$ and $\Sigma$ of Section 3). In this note a different definition of BAN estimates is made which does not require differentiability, at the same time relaxing the conditions on $\zeta$ and $\Sigma,$ whereas in essence all important theorems in the theory of BAN estimates are retained.