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 $a^*\hat{C}_N(\rho)^{-1}b$, where $a,b\in{\mathbb C}^N$ are unit norm deterministic vectors and $\hat{C}_N(\rho)$ a robust-shrinkage estimator of scatter parametrized by $\rho$ and built upon $n$ independent elliptical vector observations, is presented. The fluctuations of $a^*\hat{C}_N(\rho)^{-1}b$ are found to be of order $N^{-\frac12}$ and to be the same as those of $a^*\hat{S}_N(\rho)^{-1}b$ for $\hat{S}_N(\rho)$ 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 $\rho$.