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Optimal bounds for aggregation of affine estimators

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  • Pierre Bellec

    (CREST, ENSAE, UMR CNRS 9194)

Abstract

This paper deals with aggregation of estimators in the context fixed design regression, with heteroscedastic and subgaussian noise. We derive sharp oracle inequalities in deviation for model selection type aggregation of affine estimators when the noise is subgaussian. Explicit numerical constants are given for Gaussian noise and the procedure is robust to variance misspecification. Then we present a new concentration result that is sharper than the Hanson-Wright inequality under the Bernstein condition on the noise. This allows us to improve the sharp oracle inequality obtained in the subgaussian case. Finally, we show that up to numerical constants, the optimal sparsity oracle inequality previously obtained for Gaussian noise holds in the subgaussian case. The exact knowledge of the variance of the noise is not needed to construct the estimator that satisfies the sparsity oracle inequality.

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  • Pierre Bellec, 2015. "Optimal bounds for aggregation of affine estimators," Working Papers 2015-06, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2015-06
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    References listed on IDEAS

    as
    1. H. Dette & A. Munk & T. Wagner, 1998. "Estimating the variance in nonparametric regression—what is a reasonable choice?," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 751-764.
    2. Axel Munk & Nicolai Bissantz & Thorsten Wagner & Gudrun Freitag, 2005. "On difference‐based variance estimation in nonparametric regression when the covariate is high dimensional," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 19-41, February.
    3. Tingni Sun & Cun-Hui Zhang, 2012. "Scaled sparse linear regression," Biometrika, Biometrika Trust, vol. 99(4), pages 879-898.
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