Pivotal estimation via square-root lasso in nonparametric regression
We propose a self-tuning âˆš Lasso method that simultaneiously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity, and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in contrast to Lasso. We establish various non-asymptotic bounds for âˆš Lasso including prediction norm rate and sharp sparcity bound. Our analysis is based on new impact factors that are tailored to establish prediction rates. In order to cover heteroscedastic non-Gaussian noise, we rely on moderate deviation theory for self-normalized sums to achieve Gaussian-like results under weak conditions. Moreover, we derive bounds on the performance of ordinary least square (ols) applied to the model selected by âˆš Lasso accounting for possible misspecification of the selected model. Under mild conditions the rate of convergence of ols post âˆš Lasso is no worse than âˆš Lasso even with a misspecified selected model and possibly better otherwise. As an application, we consider the use of âˆš Lasso and post âˆš Lasso as estimators of nuisance parameters in a generic semi-parametric problem (nonlinear instrumental/moment condition or Z-estimation problem).
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- 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.
- Amemiya, Takeshi, 1977. "The Maximum Likelihood and the Nonlinear Three-Stage Least Squares Estimator in the General Nonlinear Simultaneous Equation Model," Econometrica, Econometric Society, vol. 45(4), pages 955-68, May.
- repec:cup:cbooks:9780521496032 is not listed on IDEAS
- 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.
- 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.
- Hans-Ulrich Derlien & B. Guy Peters, 2008. "Introduction," Chapters, in: The State at Work, Volume 2, chapter 1 Edward Elgar.
- Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-54, July.
- Tingni Sun & Cun-Hui Zhang, 2012. "Scaled sparse linear regression," Biometrika, Biometrika Trust, vol. 99(4), pages 879-898.
- Eric Gautier & Alexandre Tsybakov, 2011.
"High-Dimensional Instrumental Variables Regression and Confidence Sets,"
2011-13, Centre de Recherche en Economie et Statistique.
- Eric Gautier & Alexandre Tsybakov, 2014. "High-dimensional instrumental variables regression and confidence sets," Working Papers hal-00591732, HAL.
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