Robust minimax detection of a weak signal in noise with a bounded variance and density value at the center of symmetry

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In practical communication environments, it is frequently observed that the underlying noise distribution is not Gaussian and may vary in a wide range from short-tailed to heavy-tailed forms. To describe partially known noise distribution densities, a distribution class characterized by the upper-bounds upon a noise variance and a density dispersion in the central part is used. The results on the minimax variance estimation in the Huber sense are applied to the problem of asymptotically minimax detection of a weak signal. The least favorable density minimizing Fisher information over this class is called the Weber-Hermite density and it has the Gaussian and Laplace densities as limiting cases. The subsequent minimax detector has the following form: i) with relatively small variances, it is the minimum L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm distance rule; ii) with relatively large variances, it is the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 </sub> -norm distance rule; iii) it is a compromise between these extremes with relatively moderate variances. It is shown that the proposed minimax detector is robust and close to Huber's for heavy-tailed distributions and more efficient than Huber's for short-tailed ones both in asymptotics and on finite samples