An example of non-standard behavior of a maximum likelihood estimator in the large sample regime

In this paper, the performance of a maximum likelihood estimator (MLE) for a signal model accounting for possible coherence of the signal sources is studied. It is done by combining a dynamical evolution model of the source amplitude, namely a Gaussian random walk, with the usual observation model, assumed to be Gaussian. This implies an underlying relation between the signal source parameters and both the mean and the covariance matrix of the observations, and hence leads to a more general model than those usually used in array processing, such as the conditional signal model or the unconditional one. The resulting so-called "generalized conditional MLE" can be expressed both in batch form and in recursive form, which is updated since a term was missing in a previous paper. In addition, we derive the Cramér-Rao bounds associated with a model of this type, accounting for the fluctuations of a radar target's backscattering coefficient. Simulation results highlight a non-standard behavior of the estimators: consistent but not efficient for frequencies, efficient but non-consistent for amplitudes.