The Other Side of the Lower Bound. A Note with a Correction

Abstract

Abstract This note has as its primary purpose the correction of a statement made in a previous paper regarding the attainment of the variance of a proposed estimator of its lower bound. At the same time it seemed useful, as an expository function, to recapitulate a development of the lower bound which led this writer to a better understanding of it, and to the realization of the indicated error. Here the bound is developed ab initio from a simple identity, instead of secondarily from Schwartz's inequality, as is usual. It should be stated that, mathematically, there is nothing essentially new in this development; however, it yields, in addition to a clarification of the “why” of the bound, a necessary, condition for its attainment. The element of novelty in the statement of this condition is doubtless mathematically trivial, but may be of use in application. Also it is hoped that its direct derivation from the same elementary identity as the bound itself is derived, may make it understood by the nonmathematician—and possibly even by mathematicians! The usual estimate of the mean of the normal function, whose variance is well known to attain the lower bound, is used as an example to illustrate how this condition is fulfilled, and the logistic function illustrates how it is not fulfilled. The question dealt with—attaining of the variance of an estimator to its lower bound—is, of itself, of only limited technical interest. Its relation to broader questions of estimation, in particular, maximum likelihood estimation, cannot be dealt with in this short communication. Clearly, the fact that an estimator has lower-bound variance is no guarantee that the estimator is good, much less that it is “best.” The bound may itself be very high and besides, the bias of the estimator may be large, so that the mean square error can be enormous compared with another estimator whose Variance does not attain its much lower-bound, and has little or no bias. For the author's views on more general questions of “optimum” estimation, interested readers may wish to consult papers [1], [2], [3], [4], [5], [7].