On unbiased estimates of uniformly minimum variance

Abstract

In a given statistical framework let T be the class of all estimates that are the uniformly minimum variance estimates of their respective expected values. Let T_b denote the class of bounded estimates in T. The main conclusions of the paper may then be outlined as follows, (i) There exists a statistic such that T_b is the class of all bounded functions of this statistic; moreover, every real valued function of this statistic is in T. It follows, in particular, that if t is in T_b and u is a real valued function of t, then u is in T, (ii) T contains an unbiased estimate of every estimable parameter if and only if the framework admits a complete sufficient statistic. In this ease, as is well known, T is the class of all real valued functions of the complete sufficient statistic. Conclusion (ii) can also be stated as follows. Suppose that in the given framework the maximum possible reduction of the sample space by means of sufficient statistics has already been carried out. Then either each estimable parameter has a unique unbiased estimate, or there exist estimable parameters that do not admit unbiased estimates of uniformly minimum variance. A more precise statement and discussion of the above conclusions is deferred to later sections. The conclusions are established under the mild restriction that the sample space is, or may be taken to be, a subset of the m dimensional Cartesian space (1 ≤ m ≤ ∞), and that the alternative distributions of the sample point admit density functions with respect to a fixed σ-finite measure. It is shown by an example that the restriction to bounded estimates is essential to conclusion (i).