A theory of Cramer-Rao bounds for constrained parametric models

A simple expression for the Cramer-Rao bound (CRB) is presented for the scenario of estimating parameters theta that are required to satisfy a differentiable constraint function f( theta). A proof of this constrained CRB (CCRB) is provided using the implicit function theorem, and the encompassing theory of the CCRB is proven in a similar manner. This theory includes connecting the CCRB to notions of identifiability of constrained parameters; the linear model under a linear constraint; the constrained maximum likelihood problem, it's asymptotic properties and the method of scoring with constraints; and hypothesis testing. The value of the tools developed in this theory are then presented in the communications context for the convolutive mixture model and the calibrated array model.