  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 $\rho$.
