Second order statistics of robust estimators of scatter. Application to GLRT detection for elliptical signals

A central limit theorem for bilinear forms of the type aĈN (ρ) −1b, where a, b ∈ C are unit norm deterministic vectors and ĈN (ρ) a robust-shrinkage estimator of scatter parametrized by ρ and built upon n independent elliptical vector observations, is presented. The fluctuations of aĈN (ρ) −1b are found to be of order N− 1 2 and to be the same as those of aŜN (ρ) −1b for ŜN (ρ) a matrix of a theoretical tractable form. This result is exploited in a classical signal detection problem to provide an improved detector which is both robust to elliptical data observations (e.g., impulsive noise) and optimized across the shrinkage parameter ρ.