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Aggregation of affine estimators

Author(s): Dai, Dong; Rigollet, Philippe; Xia, Lucy; Zhang, Tong

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Abstract: We consider the problem of aggregating a general collection of affine estimators for fixed design regression. Relevant examples include some commonly used statistical estimators such as least squares, ridge and robust least squares estimators. Dalalyan and Salmon [DS12] have established that, for this problem, exponentially weighted (EW) model selection aggregation leads to sharp oracle inequalities in expectation, but similar bounds in deviation were not previously known. While results [DRZ12] indicate that the same aggregation scheme may not satisfy sharp oracle inequalities with high probability, we prove that a weaker notion of oracle inequality for EW that holds with high probability. Moreover, using a generalization of the newly introduced Q-aggregation scheme we also prove sharp oracle inequalities that hold with high probability. Finally, we apply our results to universal aggregation and show that our proposed estimator leads simultaneously to all the best known bounds for aggregation, including ℓq-aggregation, q ∈ (0, 1), with high probability.
Publication Date: 2014
Citation: Dai, Dong, Rigollet, Philippe, Xia, Lucy, Zhang, Tong. (2014). Aggregation of affine estimators. Electronic Journal of Statistics, 8 (1), 302 - 327. doi:10.1214/14-EJS886
DOI: doi:10.1214/14-EJS886
EISSN: 1935-7524
Pages: 302 - 327
Type of Material: Journal Article
Journal/Proceeding Title: Electronic Journal of Statistics
Version: Final published version. This is an open access article.

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