Concerning some inequalities in the theory of statistical estimation

Abstract

Abstract 1. Introduction. This note is centered around nn alternative derivation of CRAMÉR'S. inequality ([1], p. 93, (17)) relating to the covariance matrix of an unbiased estimate of a k-dimensional parameter point. Certain important differences place this derivation in contrast to that in (1). In the first place it will not be necessary for us to make use of conditional probability distributions. and consequently we are able to allow complete generality to the structure of the sample space. Secondly, whereas in [l] it is assumed that the covariance matrices of the estimate are non-singular, and it then follows that the matrices (cf. (7) below) are non-singular, we on the other hand have these two non-singularity assertions reversed in the rOles of premise and conclusion. That is, we assume the non-singularity of and find as a consequence that tbe covariance matrices of the estimate are non-singular. To make fewer assumptions concerning the estimate seems a little more natural in the present context. However, when we consider that the non-singularity of the matrices O ε Θ = an open set in Euclidean k-space, is equivalent to a certain type of k-dimensionality of the family β = }p θ,Θ ε Θ } of probability densities nnder consideration ([2J; cf. Abstract, Annals of Math. Stat., Vol. XIX, No.1, March, 1(48), then we see that these two points of view give significant, complementary results. Our finding is that if β is k-dimensional then a finite-variance, unbiased estimate has non-singular covariance matrices. CRAMER'S corollary would be: if there exists a finite-variance, unbiased estimate with non-singular covariance matrices then β is k-dimensional.