In this paper, we deal with covariance matrix estimation in complex elliptically symmetric (CES) distributions. We focus on Tyler's estimator (TyE) and the well-known sample covariance matrix (SCM). TyE is widely used in practice, but its statistical behavior is still poorly understood. On the other hand, under Gaussian assumption, the SCM is Wishart-distributed, but its properties degrade in non-Gaussian environments. The main contribution is the derivation of new properties of TyE under CES framework, in order to approximate its behavior with a simpler one, the Wishart one. Finally, Monte-Carlo simulations support that claims and demonstrate the interest of this result.

In this paper, we propose a new regularized (penalized) covariance matrix estimator which encourages grouping of the eigenvalues by penalizing large differences (gaps) between successive eigenvalues. This is referred to as fusing eigenvalues (eFusion), The proposed penalty function utilizes Tukey's biweight function that is widely used in robust statistics. The main advantage of the proposed method is that it has very small bias for sufficiently large values of penalty parameter. Hence, the method provides accurate grouping of eigenvalues. Such benefits of the proposed method are illustrated with a numerical example, where the method is shown to perform favorably compared to a state-of-art method.

The purpose of this paper is to derive new asymptotic properties of the robust adaptive normalized matched filter (ANMF). More precisely, the ANMF built with Tyler estimator (TyE-ANMF) is analyzed under the framework of complex elliptically symmetric (CES) distributions. We show that the distribution of TyE-ANMF can be accurately approximated by the well-known distribution of the ANMF built with the sample covariance matrix (SCM-ANMF) under the Gaussian assumption. To that end, the asymptotic properties of the difference between both ANMF detectors are derived. By comparison with the state of the art, the asymptotic properties of the TyE-ANMF are shown to be better approximated by the SCM-ANMF rather than using the NMF (test built with the true CM). Some Monte-Carlo simulations support that claim and demonstrate the interest of this theoretical result.

No abstract