Semiparametric Inference and Lower Bounds for Real Elliptically Symmetric Distributions

This paper has a twofold goal. The first aim is to provide a deeper understanding of the family of the Real Elliptically Symmetric (RES) distributions by investigating their intrinsic semiparametric nature. The second aim is to derive a semiparametric lower bound for the estimation of the parametric component of the model. The RES distributions represent a semiparametric model where the parametric part is given by the mean vector and by the scatter matrix while the non-parametric, infinite-dimensional, part is represented by the density generator. Since, in practical applications, we are often interested only in the estimation of the parametric component, the density generator can be considered as nuisance. The first part of the paper is dedicated to conveniently place the RES distributions in the framework of the semiparametric group models. The second part of the paper, building on the mathematical tools previously introduced, the Constrained Semiparametric Cram\'{e}r-Rao Bound (CSCRB) for the estimation of the mean vector and of the constrained scatter matrix of a RES distributed random vector is introduced. The CSCRB provides a lower bound on the Mean Squared Error (MSE) of any robust $M$-estimator of mean vector and scatter matrix when no a-priori information on the density generator is available. A closed form expression for the CSCRB is derived. Finally, in simulations, we assess the statistical efficiency of the Tyler's and Huber's scatter matrix $M$-estimators with respect to the CSCRB.Comment: This paper has been accepted for publication in IEEE Transactions on Signal Processin

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 2: The Under-Sampled Case

In the first part of this series of two papers, we extended the expected likelihood approach originally developed in the Gaussian case, to the broader class of complex elliptically symmetric (CES) distributions and complex angular central Gaussian (ACG) distributions. More precisely, we demonstrated that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual scatter matrix \mSigma_{0} does not depend on the latter: it only depends on the density generator for the CES distribution and is distribution-free in the case of ACG distributed data, i.e., it only depends on the matrix dimension $M$ and the number of independent training samples $T$, assuming that $T \geq M$. Additionally, regularized scatter matrix estimates based on the EL methodology were derived. In this second part, we consider the under-sampled scenario ($T \leq M$) which deserves a specific treatment since conventional maximum likelihood estimates do not exist. Indeed, inference about the scatter matrix can only be made in the $T$-dimensional subspace spanned by the columns of the data matrix. We extend the results derived under the Gaussian assumption to the CES and ACG class of distributions. Invariance properties of the under-sampled likelihood ratio evaluated at \mSigma_{0} are presented. Remarkably enough, in the ACG case, the p.d.f. of this LR can be written in a rather simple form as a product of beta distributed random variables. The regularized schemes derived in the first part, based on the EL principle, are extended to the under-sampled scenario and assessed through numerical simulations

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

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 Cram\'{e}r-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 \textit{Wirtinger} or $\mathbb{C}\mathbb{R}$-\textit{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 matrix of a CES-distributed vector in the presence of unknown, nuisance, density generator. Furthermore, we show how to exploit the derived SSB formula to obtain the semiparametric counterpart of the Stochastic CRB for Direction of Arrival (DOA) estimation under a random signal model assumption. Simulation results are also provided to clarify the theoretical findings and to demonstrate their usefulness in common array processing applications.Comment: Submitted to IEEE Transactions on Signal Processing. arXiv admin note: substantial text overlap with arXiv:1807.08505, arXiv:1807.0893

Contour projected dimension reduction

In regression analysis, we employ contour projection (CP) to develop a new dimension reduction theory. Accordingly, we introduce the notions of the central contour subspace and generalized contour subspace. We show that both of their structural dimensions are no larger than that of the central subspace Cook [Regression Graphics (1998b) Wiley]. Furthermore, we employ CP-sliced inverse regression, CP-sliced average variance estimation and CP-directional regression to estimate the generalized contour subspace, and we subsequently obtain their theoretical properties. Monte Carlo studies demonstrate that the three CP-based dimension reduction methods outperform their corresponding non-CP approaches when the predictors have heavy-tailed elliptical distributions. An empirical example is also presented to illustrate the usefulness of the CP method.Comment: Published in at http://dx.doi.org/10.1214/08-AOS679 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 1: The Over-Sampled Case

In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix $\R_{0}$ does not depend on this matrix and is fully specified by the matrix dimension $M$ and the number of independent training samples $T$. Since this p.d.f. could therefore be pre-calculated for any a priori known $(M,T)$, one gets a possibility to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically ``as likely'' as the a priori unknown actual covariance matrix. This ``expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called ``threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative of CES, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix \mSigma_{0}. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario ($T \geq M$) while Part 2 deals with the under-sampled scenario ($T \leq M$)

Asymptotic distributions of robust shape matrices and scales

It has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix ∑ of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for ∑ or another matrix Γ ∝ ∑. In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the presented estimators are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the presented scale function leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine (2008) in the context of local asymptotic normality theory. --local asymptotic normality,M-estimator,R-estimator,robust covariance matrix estimator,scale-invariant function,S-estimator,shape matrix,Tyler's M-estimator

Generalized robust shrinkage estimator and its application to STAP detection problem

Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tyler's estimator [1] also called the Fixed-Point Estimator (FPE) [2], a "shrinkage" fixed-point estimator has been introduced in [3]. First, this work extends the results of [3,4] by giving the general solution of the "shrinkage" fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter $\beta$ (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data