On constrained and regularized high-dimensional regression

Annals of the Institute of Statistical Mathematics

Volumn 65, Issue 5, 2013, Pages 807-832

  Shen, Xiaotong ^a   Pan, Wei ^a   Zhu, Yunzhang ^a   Zhou, Hui ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

(p, n) versus fixed p asymptotics; Constrained regression; Difference convex programming; Nonconvex regularization; Parameter and nonparametric models

Indexed keywords

(P, N) VERSUS FIXED P-ASYMPTOTICS; CONSTRAINED REGRESSION; DIFFERENCE CONVEX PROGRAMMING; NON-PARAMETRIC MODEL; NONCONVEX;

MATHEMATICAL TECHNIQUES; STATISTICS;

SEPARATION;

EID: 84885310442     PISSN: 00203157     EISSN: 15729052     Source Type: Journal    
DOI: 10.1007/s10463-012-0396-3     Document Type: Article

References (25)

1. Akaike, H. (1973). Information theory and the maximum likelihood principle. In V. Petrov, F. Csáki (Eds.), International symposium on information theory (pp. 267-281). Budapest: Akademiai Kiádo.
2. Chen, J.; Chen, Z. (2008). Extended Bayesian information criterion for model selection with large model space. Biometrika, 95, 759-771.
3. Chen, S.S.; Donoho, D.; Saunders, M.A. (2001). Atomic decomposition by basis pursuit. SIAM Review, 43, 129-159.
4. Fan, J.; Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96, 1348-1360.
5. Gu, C. (1998). Model indexing and smoothing parameter in nonparametric function estimation. Statistica Sinica, 8, 607-646.
6. Ibragimov, I.; Has'minskii, R. (1981). Statistical estimation. New York: Springer.
7. Judge, G.G.; Bock, M.E. (1978). The statistical implications of pretest and Stein-rule estimators in econometrics. Amsterdam: North-Holland.
8. Kim, Y.; Choi, H.; Oh, H.-S. (2008). Smoothly clipped absolute deviation of high dimensions. Journal of the American Statistical Association, 103, 1665-1673.
9. Liu, W.; Yang, Y.Y. (2010). Consistency for BIC selection (manuscript)
10. Lv, J.; Fan, Y. (2009). A unified approach to model selection and sparse recovery using regularized least squares. The Annals of Statistics, 37, 3498-3528.
11. Meinshausen, N.; Buhlmann, P. (2006). High dimensional graphs and variable selection with the Lasso. The Annals of Statistics, 34, 1436-1462.
12. Osborne, M.R.; Presnell, B.; Turlach, B.A. (2000). On the Lasso and its dual. Journal of Computational and Graphical Statistics, 9, 319-337.
13. Rainaldo, A. (2007). A note on the uniqueness of the Lasso solution. Technical Report, Department of Statistics, Carnegie Mellon University.
14. Raskutti, G.; Wainwright, M.; Yu, B. (2009). Minimax rates of estimation for high-dimensional linear regression over l-q balls. Technical Report, UC Berkeley.
15. Rockafellar, R.T.; Wets, R.J.B. (2011). Variational analysis. vol 317. New York: Springer.
16. Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461-464.
17. Shen, X.; Pan, W.; Zhu, Y. (2012). Likelihood-based selection and sharp parameter estimation. Journal of the American Statistical Association, 107, 223-232.
18. Tibshirani, R. (1996). Regression shrinkage and selection via the LASSO. Journal of the Royal Statistical Society, Series B, 58, 267-288.
19. Wainwright, M. (2009). Sharp thresholds for high-dimensional and noisy sparsity recovery of sparsity. IEEE Transactions on Information Theory, 55, 2183-2202.
20. Yang, Y.; Barron, A. (1998). An asymptotic property of model selection criteria. IEEE Transactions on Information Theory, 44, 95-116.
21. Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics, 38, 894-942.
22. Zhao, P.; Yu, B. (2006). On model selection consistency of Lasso. Journal of Machine Learning Research, 7, 2541-2563.
23. Zhou, S.; Shen, X.; Wolfe, D. (1998). Local asymptotics for regression splines and confidence regions. The Annals of Statistics, 26, 1760-1782.
24. Zou, H. (2006). The adaptive Lasso and its oracle properties. Journal of the American Statistical Association, 101, 1418-1429.
25. Zou, H.; Li, R. (2008). One-step sparse estimates in nonconcave penalized likelihood models. The Annals of Statistics, 36, 1509-1533.