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Asymptotic linear expansion of regularized M-estimators

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  • Tino Werner

    (Carl von Ossietzky University Oldenburg)

Abstract

Parametric high-dimensional regression requires regularization terms to get interpretable models. The respective estimators correspond to regularized M-functionals which are naturally highly nonlinear. Their Gâteaux derivative, i.e., their influence curve linearizes the asymptotic bias of the estimator, but only up to a remainder term which is not guaranteed to tend (sufficiently fast) to zero uniformly on suitable tangent sets without profound arguments. We fill this gap by studying, in a unified framework, under which conditions the M-functionals corresponding to convex penalties as regularization are compactly differentiable, so that the estimators admit an asymptotically linear expansion. This key ingredient allows influence curves to reasonably enter model diagnosis and enable a fast, valid update formula, just requiring an evaluation of the corresponding influence curve at new data points. Moreover, this paves the way for optimally-robust estimators, bounding the influence curves in a suitable way.

Suggested Citation

  • Tino Werner, 2022. "Asymptotic linear expansion of regularized M-estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(1), pages 167-194, February.
    • Handle: RePEc:spr:aistmt:v:74:y:2022:i:1:d:10.1007_s10463-021-00792-5
    DOI: 10.1007/s10463-021-00792-5
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    References listed on IDEAS

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    1. Pötscher, Benedikt M. & Leeb, Hannes, 2009. "On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2065-2082, October.
    2. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, January.
    3. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    4. Helmut Rieder & Matthias Kohl & Peter Ruckdeschel, 2008. "The cost of not knowing the radius," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 17(1), pages 13-40, February.
    5. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    6. Sara Geer, 2014. "Weakly decomposable regularization penalties and structured sparsity," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 72-86, March.
    7. Pupashenko, Daria & Ruckdeschel, Peter & Kohl, Matthias, 2015. "L2 differentiability of generalized linear models," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 155-164.
    8. Beutner, Eric & Zähle, Henryk, 2010. "A modified functional delta method and its application to the estimation of risk functionals," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2452-2463, November.
    9. Hable, Robert, 2012. "Asymptotic normality of support vector machine variants and other regularized kernel methods," Journal of Multivariate Analysis, Elsevier, vol. 106(C), pages 92-117.
    10. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2012. "Qualitative and infinitesimal robustness of tail-dependent statistical functionals," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 35-47, January.
    11. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    12. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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