On the Attainment of the Cramer-Rao Lower Bound

in which po is the density of the random variable X. It is usually stated that the lower bound can be attained only if the family of distributions of X is oneparameter exponential [2], [3] and [7, page 187]. That this is to be expected can be seen by realizing that (1) is nothing else but a statement that the square of the correlation between t(X) and (a/la) log p,(X) cannot exceed 1, and equality is attained if and only if (a/la) log p0(x) = a(O)t(x) + b(6) for some functions a and b. Then by integration over 6 the desired exponential form of p, is obtained. However, this heuristic approach conveniently ignores the fact that the above affine relation between t(x) and (a/aO) logp0(x) may fail to hold on a null set which may depend on 0 and that a priori nothing can be assumed about the functions a and b, not even measurability, let alone integrability. Since a rigorous proof does not seem to have appeared in the literature it may be appropriate to produce one here, the more so since the proof seems to be neither completely trivial nor standard. The following assumptions will be made. The sample space is an arbitrary measure space (2, X, ,), with , sigma-finite. The parameter space is the measure space (9, , 4), with e a Borel subset of the real line, X the Borel subsets of e and v Lebesgue measure. There is given a random variable X with values in z2' and distribution P0(dx) = p,(x)pe(dx), 6 e 9. For convenience differentiation with respect to a will be denoted by D. Any integration with respect to ft will always be understood to be over the whole of 2t. We shall make the following.