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