Oracle Inequalities for Convex Loss Functions with Non-Linear Targets
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.
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Jianqing Fan & Jinchi Lv & Lei Qi, 2011. "Sparse High-Dimensional Models in Economics," Annual Review of Economics, Annual Reviews, vol. 3(1), pages 291-317, 09.
- Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
- Kock, Anders Bredahl, 2013.
"Oracle Efficient Variable Selection In Random And Fixed Effects Panel Data Models,"
Cambridge University Press, vol. 29(01), pages 115-152, February.
- Anders Bredahl Kock, 2010. "Oracle Efficient Variable Selection in Random and Fixed Effects Panel Data Models," CREATES Research Papers 2010-56, Department of Economics and Business Economics, Aarhus University.
- 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 13-062, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 21 Oct 2013.
- 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.
- A. Belloni & D. Chen & V. Chernozhukov & C. Hansen, 2012.
"Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain,"
Econometric Society, vol. 80(6), pages 2369-2429, November.
- Alexandre Belloni & D. Chen & Victor Chernozhukov & Christian Hansen, 2010. "Sparse models and methods for optimal instruments with an application to eminent domain," CeMMAP working papers CWP31/10, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
- Kock, Anders Bredahl & Callot, Laurent, 2015.
"Oracle inequalities for high dimensional vector autoregressions,"
Journal of Econometrics,
Elsevier, vol. 186(2), pages 325-344.
- Anders Bredahl Kock & Laurent A.F. Callot, 2012. "Oracle Inequalities for High Dimensional Vector Autoregressions," CREATES Research Papers 2012-16, Department of Economics and Business Economics, Aarhus University.
- 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.
- Jianqing Fan & Jinchi Lv, 2008. "Sure independence screening for ultrahigh dimensional feature space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(5), pages 849-911.
- Chen, Xiaohong, 2007. "Large Sample Sieve Estimation of Semi-Nonparametric Models," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 76 Elsevier.
- A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
- 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.
- 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.
- 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, vol. 75(3), pages 531-552, 06.
When requesting a correction, please mention this item's handle: RePEc:aah:create:2013-51. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: ()
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.