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 a∗CˆN(ρ)−1b, where a,b∈CN are unit norm deterministic vectors and CˆN(ρ) a robust-shrinkage estimator of scatter parametrized by ρ and built upon n independent elliptical vector observations, is presented. The fluctuations of a∗CˆN(ρ)−1b are found to be of order N−12 and to be the same as those of a∗SˆN(ρ)−1b for SˆN(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.