Slepian-Bangs-type formulas and the related Misspecified Cramér-Rao Bounds for Complex Elliptically Symmetric distributions

Abstract

S. Fortunati). ing (MRI), have highlighted the impulsive, heavy-tailed behaviour of the observations [1] . These experimental evidences have motivated the need to go beyond the Gaussian model and develop new statistical models able to better characterize the data. One of the more flexible and general non-Gaussian model is represented by the set of the Complex Elliptically Symmetric (CES) distributions [2] , also called Multivariate Elliptically Contoured distributions [3] . CES distributions encompasses the complex Gaussian, the Generalized Gaussian and all the Compound Gaussian (CG) distributions, such as the complex t -distribution and the K -distribution, as special cases. The pdf of a CES distributed N -dimensional random vector x l ∈ C N is completely characterized by the mean value γ, the scatter (or shape) matrix and by a real valued function w (t) : R + → R , called the density generator , i.e. x l ∼ CES N ( γ, , w ) [2,3] . The CES distributions have been used in a variety of applications, in particular in the radar and array signal processing fields.Other experimental evidences reveal recurring violations of the matched model assumption, that is the claim of a perfect match between the assumed and the true data model. The mathematical bases of a formal theory of the parameter estimation under model misspecification has been firstly developed by statisticians as Huber [4] , White [5] and Vuong [6] and recently rediscovered by the Signal Processing (SP) community [7-9] and applied to a va-

Massive MIMO Radar for Target Detection

“…, x 0 }. Clearly, the joint pdf of the disturbance vector c can be obtained from (20) by setting α = 0, i.e.…”

“…A close inspection of (20) reveals that the evaluation of p X requires exact knowledge of: i) the order p of the AR model; and ii) the pdf p w of the innovations. In practice, both are rarely available.…”

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

“…Finally, by taking the expectation w.r.t. the true pdf p 0 (z) and by using the relations derived in (B.4)-(B.10) of [42,Appendix B], it is easy to verify that each entry of the SFIMĪ(θ 0 |h 0 ) can be expressed as:…”

“…Richmond and Horowitz in [35] showed an extension of the classical, Gaussian-based, SB formula to estimation problems under model misspecification. The natural follow-on [41] and [35] has been proposed in [42], where SB-type formulas, that encompass those previously obtained in [41] and [35] as special cases, have been derived for parameter estimation problems involving CES-distributed data under model misspecification. In this paper, we take a step forward to the generalization of the SB formula for semiparametric estimation in the CES framework.…”

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:…”