On the achievability of the Cramer-Rao bound for Poisson distribution

This correspondence examines the Cramer-Rao (CR) bound for data obtained in emission tomography. The likelihood function involved is the combined probability of independent Poisson random variables, the expectation of each being a linear function c_i^Tλ of the parameter vector λ. We investigated the achievability of the CR bound in the interior and on the boundary of the domain of the problem. For the former, we found that the CR bound is achievable if and only if the c_i vectors are obtained from a basis for R^N, by repeating some vectors, multiplied by constant factors. A similar result holds for the boundary case. The practical implication of the achievability condition is that the CR bound is not attainable for typical emission tomographic systems