A new multivariate Cramer-Rao inequality for parameter estimation

A new form of the Cramer-Rao inequality for the estimator of vector parameter constants is presented. For a scalar Cramer-Rao inequality of the form of the one derived, the so-called Cramer-Rao lower bound does not have a denominator that must be maximized over all components of some matrix as was required in previous multivariate derivations. For a certain class of maximum likelihood parameter estimation problems, the Cramer-Rao lower bound is the error of estimation. For this class of problems, a denominator having the form exhibited by this lower bound involves a trace and is shown to be a norm squared in a Hilbert space. Minimizing the error of estimation is shown to be equivalent to maximizing the norm in a Hilbert space while constrained to a specific compact set which represents practical constraints. The specification of input probing functions to aid in the estimation of input gain parameters in a linear dynamical system with system process noise is considered as a special case of this class of maximum likelihood parameter estimation problems. The probing functions are bang-bang.