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Oracle Inequalities for Convex Loss Functions with Non-Linear Targets

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  • Mehmet Caner

    ()
    (North Carolina State University)

  • Anders Bredahl Kock

    ()
    (Aarhus University and CREATES)

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    Abstract

    This paper consider penalized empirical loss minimization of convex loss functions with unknown non-linear target functions. Using the elastic net penalty we establish a finite sample oracle inequality which bounds the loss of our estimator from above with high probability. If the unknown target is linear this inequality also provides an upper bound of the estimation error of the estimated parameter vector. These are new results and they generalize the econometrics and statistics literature. Next, we use the non-asymptotic results to show that the excess loss of our estimator is asymptotically of the same order as that of the oracle. If the target is linear we give sufficient conditions for consistency of the estimated parameter vector. Next, we briefly discuss how a thresholded version of our estimator can be used to perform consistent variable selection. We give two examples of loss functions covered by our framework and show how penalized nonparametric series estimation is contained as a special case and provide a finite sample upper bound on the mean square error of the elastic net series estimator.

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    File URL: ftp://ftp.econ.au.dk/creates/rp/13/rp13_51.pdf
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    Bibliographic Info

    Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2013-51.

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    Length: 44
    Date of creation: 13 Dec 2013
    Date of revision:
    Handle: RePEc:aah:create:2013-51

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    Web page: http://www.econ.au.dk/afn/

    Related research

    Keywords: Empirical loss minimization; Lasso; Elastic net; Oracle inequality; Convex loss function; Nonparametric estimation; Variable selection.;

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    References

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    1. Kock, Anders Bredahl, 2013. "Oracle Efficient Variable Selection In Random And Fixed Effects Panel Data Models," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 29(01), pages 115-152, February.
    2. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    3. Xu Cheng & Zhipeng Liao, 2011. "Select the Valid and Relevant Moments: An Information-Based LASSO for GMM with Many Moments, Second Version," PIER Working Paper Archive, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania 13-062, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 21 Oct 2013.
    4. A. Belloni & D. Chen & Victor Chernozhukov & Christian Hansen, 2010. "Sparse models and methods for optimal instruments with an application to eminent domain," CeMMAP working papers, Centre for Microdata Methods and Practice, Institute for Fiscal Studies CWP31/10, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Jianqing Fan & Jinchi Lv, 2008. "Sure independence screening for ultrahigh dimensional feature space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, Royal Statistical Society, vol. 70(5), pages 849-911.
    6. Jianqing Fan & Jinchi Lv & Lei Qi, 2011. "Sparse High-Dimensional Models in Economics," Annual Review of Economics, Annual Reviews, Annual Reviews, vol. 3(1), pages 291-317, 09.
    7. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, Biometrika Trust, vol. 98(4), pages 791-806.
    8. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, Elsevier, vol. 79(1), pages 147-168, July.
    9. Chen, Xiaohong, 2007. "Large Sample Sieve Estimation of Semi-Nonparametric Models," Handbook of Econometrics, Elsevier, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 76 Elsevier.
    10. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    11. Anders Bredahl Kock & Laurent A.F. Callot, 2012. "Oracle Inequalities for High Dimensional Vector Autoregressions," CREATES Research Papers, School of Economics and Management, University of Aarhus 2012-16, School of Economics and Management, University of Aarhus.
    12. 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, Royal Statistical Society, vol. 67(5), pages 768-768.
    13. Yingying Fan & Cheng Yong Tang, 2013. "Tuning parameter selection in high dimensional penalized likelihood," Journal of the Royal Statistical Society Series B, Royal Statistical Society, Royal Statistical Society, vol. 75(3), pages 531-552, 06.
    14. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, Royal Statistical Society, vol. 67(2), pages 301-320.
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