The hybrid Cramér-Rao bound and the generalized Gaussian linear estimation problem

This paper explores the hybrid Cramer-Rao lower-bound (HCRLB) for a Gaussian generalized linear estimation problem in which some of the unknown parameters are deterministic while the other are random. In general, the HCRLB on the non-Bayesian parameters is not asymptotically tight. However, we show that for the generalized Gaussian linear estimation problem, the HCRLB of the deterministic parameters coincides with the CRLB, so it is an asymptotically tight bound. In addition, we show that the ML/MAP estimator [Van Trees and Bell, 2007] is asymptotically efficient for the non-Bayesian parameters while providing optimal estimate of the Bayesian parameters. The results are demonstrated on a signal processing example. It is shown the Hybrid estimation can increase spectral resolution if some prior knowledge is available only on a subset of the parameters.