The equality between the Stochastic CRB and its semiparametric version for elliptical data
Résumé
In this letter, the proof of the following quite surprising result is provided: if the data are elliptically distributed, the "parametric" Stochastic Cramér-Rao Bound (derived by assuming the a priori knowledge of the density generator) and its semiparametric counterpart (obtained by considering the density generator as an additional nuisance parameter) are equal. In other words, this means that, in the presence of finite-dimensional nuisance parameters (the source correlation matrix and noise power in the decomposition of the array covariance/scatter matrix), not knowing the data density generator does not lead to an additional loss of efficiency.
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