347
Views
3
CrossRef citations to date
0
Altmetric
Articles

Achieving the oracle property of OEM with nonconvex penalties

, &
Pages 28-36 | Received 07 Mar 2017, Accepted 30 Apr 2017, Published online: 19 May 2017

References

  • Bai, Z. D., & Yin, Y. Q. (1993). Limit of smallest eigenvalue of a large dimensional sample covariance matrix. Annals of Probability, 21, 1275–1294.
  • Breiman, L. (1995). Better subset regression using the nonnegative garrote. Technometrics, 37, 373–384.
  • Breheny, P., & Huang, J. (2011). Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection. Annals of Applied Statistics, 5, 232–253.
  • Bühlmann, P., & van de Geer, S. (2011). Statistics for high-dimensional data: Methods, theory and applications. Berlin: Springer.
  • Eicker, F. R. (1963). Asymptotic normality and consistency of the least squares estimators for families of linear regressions. The Annals of Mathematical Statistics, 34, 447–456.
  • Fan, J., & Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96, 1348–1360.
  • Fan, J., & Lv, J. (2008). Sure independence screening for ultrahigh dimensional feature space (with discussion). Journal of the Royal Statistical Society: Series B, 70, 849–911.
  • Fan, J., & Lv, J. (2011). Properties of non-concave penalized likelihood with NP-dimensionality. IEEE Transactions on Information Theory, 57, 5467–5484.
  • Fan, J., & Peng, H. (2004). Non-concave penalized likelihood with diverging number of parameters. Annals of Statistics, 32, 928–961.
  • Fan, J., Xue, L., & Zou, H. (2014). Strong oracle optimality of folded concave penalized estimation. Annals of Statistics, 42, 819–849.
  • Fan, Y., & Tang, C. Y. (2013). Tuning parameter selection in high dimensional penalized likelihood. Journal of the Royal Statistical Society, Series B, 75, 531–552.
  • Hoerl, A. E., & Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55–67.
  • Hsu, D., Kakade, S. M., & Zhang, T. (2012). A tail inequality for quadratic forms of sub-Gaussian random vectors. Electronic Journal of Probability, 17, 1–6.
  • Hunter, D. R., & Li, R. (2005). Variable selection using MM algorithms. Annals of Statistics, 33, 1617–1642.
  • Huo, X., & Chen, J. (2010). Complexity of penalized likelihood estimation. Journal of Statistical Computation and Simulation, 80, 747–759.
  • Huo, X., & Ni, X. L. (2007). When do stepwise algorithms meet subset selection criteria? Annals of Statistics, 35, 870–887.
  • Kim, Y., Choi, H., & Oh, H.-S. (2008). Smoothly clipped absolute deviation on high dimensions. Journal of American Statistical Association, 103, 1665–1673.
  • Lai, T. L., & Wei, C. Z. (1984). Moment inequalities with applications to regression and time series models. Inequalities in Statistics and Probability, IMS Lecture Notes-Monograph Series, 5, 165–172.
  • Lanczos, C. (1950). An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. Journal of Research of the National Bureau of Standards, 45, 255–282.
  • Loh, P. L., & Wainwright, M. J. (2013). Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima. Advances in Neural Information Processing Systems, 26, 476–484.
  • Mazumder, R., Friedman, J., & Hastie, T. (2011). SparseNet: Coordinate descent with non-convex penalties. Journal of American Statistical Association, 106, 1125–1138.
  • Meng, X. L. (2008). Discussion on one-step sparse estimates in nonconcave penalized likelihood models. Annals of Statistics, 36, 1542–1552.
  • Schifano, E. D., Strawderman, R., & Wells, M. T. (2010). Majorization–minimization algorithms for nonsmoothly penalized objective functions. Electronic Journal of Statistics, 4, 1258–1299.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B, 58, 267–288.
  • Tseng, P. (2001). Convergence of a block coordinate descent method for nondifferentiable minimization. Journal of Optimization Theory and Applications, 109, 475–494.
  • Tseng, P., & Yun, S. (2009). A coordinate gradient descent method for nonsmooth separable minimization. Programs in Mathematics, 117, 387–423.
  • Wang, H., Li, B., & Leng, C. (2009). Shrinkage tuning parameter selection with a diverging number of parameters. Journal of the Royal Statistical Society: Series B, 71, 671–683.
  • Wang, H., Li, R., & Tsai, C-L. (2007). Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika, 94, 553–568.
  • Wang, L., Kim, Y., & Li, R. (2013). Calibrating nonconvex penalized regression in ultra-high dimension. Annals of Statistics, 41, 2505–2536.
  • Wang, S., & Jia, Z. (1994). Inequalities in matrix theory ( In Chinese). Hefei: Anhui Education Press.
  • Xiong, S., Dai, B., Huling, J., & Qian, P. (2016). Orthogonalizing EM: A design-based least squares algorithm. Technometrics, 58, 285–293.
  • Xiong, S., Dai, B., & Qian, P. (2011). OEM algorithm for least squares problems (Unpublished report). Retrieved from http://arxiv.org/abs/1108.0185
  • Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Annals of Statistics, 38, 894–942.
  • Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101, 1418–1429.
  • Zou, H., & Li, R. (2008). One-step sparse estimates in nonconcave penalized likelihood models. Annals of Statistics, 36, 1509–1533.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.