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 ? C ? N ( ? ) - 1 b , where a , b ? C N are unit norm deterministic vectors and C ? N ( ? ) a robust-shrinkage estimator of scatter parametrized by ? and built upon n independent elliptical vector observations, is presented. The fluctuations of a ? C ? N ( ? ) - 1 b are found to be of order N - 1 2 and to be the same as those of a ? S ? N ( ? ) - 1 b for S ? N ( ? ) 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 ? .

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