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Finite-sample and asymptotic analysis of generalization ability with an application to penalized regression

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  • Ning Xu
  • Jian Hong
  • Timothy C. G. Fisher

Abstract

In this paper, we study the performance of extremum estimators from the perspective of generalization ability (GA): the ability of a model to predict outcomes in new samples from the same population. By adapting the classical concentration inequalities, we derive upper bounds on the empirical out-of-sample prediction errors as a function of the in-sample errors, in-sample data size, heaviness in the tails of the error distribution, and model complexity. We show that the error bounds may be used for tuning key estimation hyper-parameters, such as the number of folds $K$ in cross-validation. We also show how $K$ affects the bias-variance trade-off for cross-validation. We demonstrate that the $\mathcal{L}_2$-norm difference between penalized and the corresponding un-penalized regression estimates is directly explained by the GA of the estimates and the GA of empirical moment conditions. Lastly, we prove that all penalized regression estimates are $L_2$-consistent for both the $n \geqslant p$ and the $n

Suggested Citation

  • Ning Xu & Jian Hong & Timothy C. G. Fisher, 2016. "Finite-sample and asymptotic analysis of generalization ability with an application to penalized regression," Papers 1609.03344, arXiv.org, revised Sep 2016.
    • Handle: RePEc:arx:papers:1609.03344
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    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis

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