Distributional results for thresholding estimators in high-dimensional Gaussian regression models

Electronic Journal of Statistics

Volumn 5, Issue , 2011, Pages 1876-1934

  Pötscher, Benedikt M ^a   Schneider, Ulrike ^b  

^a UNIVERSITY OF VIENNA   (Austria)

^b VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

Adaptive lasso; Asymptotic distribution; Finite sample distribution; High dimensional model; Lasso; Oracle property; Penalized maximum likelihood; Thresholding; Uniform convergence rate; Variable selection; Variance estimation

Indexed keywords

EID: 84859837258     PISSN: 19357524     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-EJS659     Document Type: Article

References (21)

1. Alliney, S. and Ruzinsky, A. (1994). An Algorithm For the Minimization of Mixed l1 and l2 Norms With Applications to Bayesian Estimation. IEEE Transactions on Signal Processing 42 618-627.
2. Bauer, P., Pötscher, B. M. and Hackl, P. (1988). Model Selection by Multiple Test Procedures. Statistics 19 39-44.
3. Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet Shrinkage: Asymptopia? With discussion and a reply by the authors. Journal of the Royal Statistical Society Series B 57 301-369.
4. Fan, J. and Li, R. (2001). Variable Selection Via Nonconcave Penalized Likelihood and Its Oracle Properties. Journal of the American Statistical Association 96 1348-1360.
5. Fan, J. and Peng, H. (2004). Nonconcave Penalized Likelihood With a Diverging Number of Parameters. Annals of Statistics 32 928-961.
6. Feller, W. (1957). An Introduction to Probability Theory and Its Applications 1. Wiley, New York.
7. Frank, I. E. and Friedman, J. H. (1993). A Statistical View of Some Chemometrics Regression Tools (with discussion). Technometrics 35 109-148.
8. Ibragimov, I. A. (1956). On the Composition of Unimodal Distributions. Theory of Probability and its Applications 1 255-260.
9. Knight, K. and Fu W. (2000). Asymptotics of Lasso-Type Estimators. Annals of Statistics 28 1356-1378.
10. Leeb, H. and Pötscher, B. M. (2003). The Finite-Sample Distribution of Post-Model-Selection Estimators and Uniform Versus Nonuniform Approximations. Econometric Theory 19 100-142.
11. Leeb, H. and Pötscher, B. M. (2005). Model Selection and Inference: Facts and Fiction. Econometric Theory 21 21-59.
12. Leeb, H. and Pötscher, B. M. (2008). Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator. Journal of Econometrics 142 201-211.
13. Pötscher, B. M. (1991). Effects of Model Selection on Inference. Econometric Theory 7 163-185.
14. Pötscher, B. M. (2006). The Distribution of Model Averaging Estimators and an Impossibility Result Regarding its Estimation. IMS Lecture Notes-Monograph Series 52 113-129.
15. Pötscher, B. M. and Leeb, H. (2009). On the Distribution of Penalized Maximum Likelihood Estimators: The LASSO, SCAD, and Thresholding. Journal of Multivariate Analysis 100 2065-2082.
16. Pötscher, B. M. and Schneider, U. (2009). On the Distribution of the Adaptive LASSO Estimator. Journal of Statistical Planning and Inference 139 2775-2790.
17. Pötscher, B. M. and Schneider, U. (2010). Confidence Sets Based on Penalized Maximum Likelihood Estimators in Gaussian Regression. Electronic Journal of Statistics 4 334-360.
18. Sen, P. K. (1979). Asymptotic Properties of Maximum Likelihood Estimators Based on Conditional Specification. Annals of Statistics 7 1019-1033.
19. Tibshirani, R. (1996). Regression Shrinkage and Selection Via the Lasso. Journal of the Royal Statistical Society Series B 58 267-288.
20. Zhang, C. (2010). Nearly Unbiased Variable Selection under Minimax Concave Penalty. Annals of Statistics 38 893-942.
21. Zou, H. (2006). The Adaptive Lasso and Its Oracle Properties. Journal of the American Statistical Association 101 1418-1429.