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Large adaptive estimation in linear regression model. I: Consistency. (English) Zbl 0792.62033
Summary: A condition of identifiability of linear regression models with symmetric distribution of errors is given. Following R. Beran’s approach [Ann. Stat. 6, 292-313 (1978; Zbl 0378.62051)] for the location case, consistency and asymptotic normality of this adaptive estimator are proved. The result shows that the estimator is not asymptotically efficient. But it selects a model with such distribution function of errors which is (in the sense of Hellinger distance applied to \(F(x)\) and \(1-P(-x))\) “as much as possible symmetric”, which may be useful when we know that there are no reasons for the asymmetry.

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62G35 Nonparametric robustness
62F35 Robustness and adaptive procedures (parametric inference)
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