Constrained hypothesis testing and the Cramér-Rao bound

The classical Wald and Rao test statistics are asymptotically equivalent to the generalized likelihood ratio test statistics, while not requiring parameter estimation under both hypotheses, and so they provide lower complexity test statistics. In this paper we develop corresponding variations of the Wald and Rao test for nested hypothesis testing under parameter constraints. The resulting tests incorporate the constrained Cramér-Rao bound formulation from Stoica and Ng, and unify some asymptotic hypothesis testing results. Examples will illustrate key ideas and test performance.