Semiparametric CRB and Slepian-Bangs Formulas for Complex Elliptically Symmetric Distributions

Abstract

The main aim of this paper is to extend the semiparametric inference methodology, recently investigated for Real Elliptically Symmetric (RES) distributions, to Complex Elliptically Symmetric (CES) distributions. The generalization to the complex field is of fundamental importance in all practical applications that exploit the complex representation of the acquired data. Moreover, the CES distributions has been widely recognized as a valuable and general model to statistically describe the non-Gaussian behaviour of datasets originated from a wide variety of physical measurement processes. The paper is divided in two parts. In the first part, a closed form expression of the constrained Semiparametric Cramér-Rao Bound (CSCRB) for the joint estimation of complex mean vector and complex scatter matrix of a set of CES-distributed random vectors is obtained by exploiting the so-called Wirtinger or CR-calculus.The second part deals with the derivation of the semiparametric version of the Slepian-Bangs formula in the context of the CES model. Specifically, the proposed Semiparametric Slepian-Bangs (SSB) formula provides us with a useful and ready-to-use expression of the Semiparametric Fisher Information Matrix (SFIM) for the estimation of a parameter vector parametrizing the complex mean and the complex scatter S. Fortunati is with

Semiparametric Stochastic CRB for DOA Estimation in Elliptical Data Model

“…From In order to remove the well-known scale ambiguity, we impose a constraint on the functional form of the density generator h. Following the same procedure adopted in [9], [10], we assume that h ∈ G is parameterized in order to satisfy the constraint:…”

“…B. The Semiparametric Stochastic CRB Following the theoretical results obtained in our recent work [9], we can take these two steps further: 1) We drop the Gaussianity assumption in favour of the more general CES model in (6), 2) By relying on the semiparametric framework 2 , we consider as additional nuisace parameter the density generator h 0 itself.…”

“…Roughly speaking, the semiparametric framework addressed in [9], [14], [15] allow us to derive a lower bound on the performance of any estimator of the vector of the spatial frequencies ν 0 when the signal covariance ζ 0 , the noise power σ 2 0 and even the density generator h 0 are unknown. The only assumption used to derive this bound, that we call Semiparametric SCRB (SSCRB), is that the data snapshots are CES-distributed as in (6).…”

“…The only assumption used to derive this bound, that we call Semiparametric SCRB (SSCRB), is that the data snapshots are CES-distributed as in (6). As proved in [9], the SSCRB can be expressed as:…”

“…The reader that is not familiar with the semiparametric theory may have a look at the books[12] and[13] and to the wide statistical literature available on this topic. Moreover, we may suggest the reader to look into our recent works[9],[14],[15] where the semiparmaetric nature of the CES distributions has been analysed.…”