A few theoretical results for Laplace and arctan penalized ordinary least squares linear regression estimators

References

• Breiman, L. 1996. Heuristics of instability and stabilization in model selection. The Annals of Statistics 24 (6):2350–83. doi: 10.1214/aos/1032181158.
   Web of Science ®Google Scholar
• Candes, E. J., M. B. Wakin, and S. P. Boyd. 2008. Enhancing sparsity by reweighted l1 minimization. The Journal of Fourier Analysis and Applications 14:877–905. doi: 10.1007/s00041-008-9045-x.
   Web of Science ®Google Scholar
• Fan, J., and R. Li. 2001. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96:1348–60. doi: 10.1198/016214501753382273.
   Web of Science ®Google Scholar
• Frank, I., and J. Friedman. 1993. A statistical view of some chemometrics regression tools. Technometrics 35:109–48. doi: 10.1080/00401706.1993.10485033.
   Web of Science ®Google Scholar
• Fu, W. J. 1998. Penalized regressions: the Bridge versus the Lasso. The Journal of Computational and Graphical Statistics 7:397–416. doi: 10.2307/1390712.
   Web of Science ®Google Scholar
• Härdle, W. K., and L. Simar. 2019. Applied multivariate statistical analysis. 5th ed. Berlin-Heidelberg-New York: Springer Verlag.
   Google Scholar
• Hoerl, A. E., and R. Kennard. 1970. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12:55–67. doi: 10.1080/00401706.1970.10488634.
   Web of Science ®Google Scholar
• Huang, J., J. L. Horowitz, and S. Ma. 2008. Asymptotic properties of bridge estimators in sparse high-dimensional regression models. The Annals of Statistics 36 (2):587–613. doi: 10.1214/009053607000000875.
   Web of Science ®Google Scholar
• Kim, J., and D. Pollard. 1990. Cube root asymptotics. The Annals of Statistics 18:191–219. doi: 10.1214/aos/1176347498.
   Web of Science ®Google Scholar
• Knight, K., and W. Fu. 2000. Asymptotics for Lasso-type estimators. Annals of Statistics 28 (5):1356–78.
   Web of Science ®Google Scholar
• Loh, P., and M. J. Wainwright. 2015. Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima. Journal of Machine Learning Research 16:559–616.
   Web of Science ®Google Scholar
• Pinelis, I. 2015. Relationship between the first four moments. The American Mathematical Monthly 122 (5):479–81.
   Web of Science ®Google Scholar
• Pollard, D. 1991. Asymptotics for least absolute deviation regression estimators. Econometric Theory 7:186–99. doi: 10.1017/S0266466600004394.
   Web of Science ®Google Scholar
• Ross, N. 2011. Fundamentals of Stein’s method. Probability Surveys 8:210–93. doi: 10.1214/11-PS182.
   Google Scholar
• Rudin, W. 1976. Principles of mathematical analysis. 3rd ed. New York: McGraw-Hill.
   Google Scholar
• Tibshirani, R. 1996. Regression shrinkage and selection via the lasso. The Journal of the Royal Statistical Society, Series B 58 (1): 267–88. doi: 10.1111/j.2517-6161.1996.tb02080.x.
   Google Scholar
• Trzasko, J., and., A. Manduca. 2009. Highly undersampled magnetic resonance image reconstruction via homotopic L0-minimization. IEEE Transactions on Medical Imaging 28 (1):106–21. doi: 10.1109/TMI.2008.927346.
   PubMed Web of Science ®Google Scholar
• Van Der Vaart, A. W. 1998. Asymptotic statistics. New York: Cambridge University Press.
   Google Scholar
• Van Der Vaart, A. W., and J. A. Wellner. 1996. Weak convergence and empirical processes: With applications to statistics. New York: Springer.
   Google Scholar
• Vettam, S., and M. John. 2019. Regularized deep learning with nonconvex penalties. arXiv preprint, doi: arXiv:1909.05142.
   Google Scholar
• Vettam, S., and M. John. 2022. On two recent nonconvex penalties for regularization in machine learning. Results in Applied Mathematics 14:100256. doi: 10.1016/j.rinam.2022.100256.
   Google Scholar
• Wang, Y., and L. Zhu. 2016. Variable selection and parameter estimation with the Atan regularization method. Journal of Probability and Statistics 2016:1–12. doi: 10.1155/2016/6495417.
   Google Scholar
• Zhang, C. H. 2010. Nearly unbiased variable selection under minimax concave penalty. Annals of Statistics 38 (2):894–942.
   Web of Science ®Google Scholar
• Zou, H, and T. Hastie. 2005. Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society, Series B 67:301–20. doi: 10.1111/j.1467-9868.2005.00503.x.
   Google Scholar