Huber's Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

A brief survey of former and recent results on Hubers minimax approach inrobust statistics is given. The least informative distributions minimizing Fisher information forlocation over several distribution classes with upper-bounded variances and subranges are writtendown. These least informative distributions are qualitatively different from classical Huberssolution and have the following common structure: (i) with relatively small variances they areshort-tailed, in particular normal; (ii) with relatively large variances they are heavy-tailed, inparticular the Laplace; (iii) they are compromise with relatively moderate variances. These resultsallow to raise the efficiency of minimax robust procedures retaining high stability as compared toclassical Hubers procedure for contaminated normal populations. In application to signal detectionproblems, the proposed minimax detection rule has proved to be robust and close to Hubers forheavy-tailed distributions and more efficient than Hubers for short-tailed ones both in asymptoticsand on finite samples.