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A new perspective on robust $$M$$-estimation: finite sample theory and applications to dependence-adjusted multiple testing. (English) Zbl 1409.62154
Consider an ordinary linear regression model with a vector of regression coefficients, and the random noise variable with mean zero and finite variance. When the normality assumption is violated, robust alternatives to the method of least square, typified by the Huber estimator, are sorely needed.
In this paper, the Huber estimator with tuning parameter adapted to sample size, dimension and variance of the noise is considered. The Berry-Essen inequality and Crámer-type moderate deviation are developed, too.
As a special case, a sub-Gaussian type deviation inequality and a non-asymptotic Bahadur representation when noise variables only have second moments are established.

##### MSC:
 62J15 Paired and multiple comparisons; multiple testing 62H15 Hypothesis testing in multivariate analysis 62F35 Robustness and adaptive procedures (parametric inference) 62J05 Linear regression; mixed models 62E20 Asymptotic distribution theory in statistics
FAMT
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