Recently, a new adaptive scheme [1], [2] has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non Gaussian noise. This latter has been modelled by compound-Gaussian noise, which is the product c of the square root of a positive unknown variable τ (deterministic or random) and an independent Gaussian vector x, c = √ τ x. Because of the implicit algebraic structure of the equation to solve, we called the corresponding solution, the Fixed Point (FP) estimate. When τ is assumed deterministic and unknown, the FP is the exact Maximum Likelihood (ML) estimate of the noise covariance structure, while when τ is a positive random variable, the FP is an Approximate Maximum Likelihood (AML). This estimate has been already used for its excellent statistical properties without proofs of its existence and uniqueness. The major contribution of this paper is to fill these gaps. Our derivation is based on some Likelihood functions general properties like homogeneity and can be easily adapted to other recursive contexts. Moreover, the corresponding iterative algorithm used for the FP estimate practical determination is also analyzed and we show the convergence of this recursive scheme, ensured whatever the initialization.

Abstract-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 FixedPoint Estimator (FPE) [2], a "shrinkage" fixed-point estimator has been originally introduced in [3]. First, this work extends the results of [4], [5] 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 β (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.

No abstract