Non-Convex Penalized Estimation of Count Data Responses via Generalized Linear Model (GLM)


  • Rasaki Olawale Olanrewaju Department of Statistics, University of Ibadan, Ibadan, Nigeria
  • Johnson Funminiyi Ojo Department of Statistics, University of Ibadan, Ibadan, Nigeria



Count Data, Minimax Concave Penalty (MCP), Non-convex penalization, Smoothed Clipped Absolute Deviation


This study provided a non-convex penalized estimation procedure via Smoothed Clipped Absolute Deviation (SCAD) and Minimax Concave Penalty (MCP) for count data responses to checkmate the problem of covariates exceeding the sample size . The Generalized Linear Model (GLM) approach was adopted in obtaining the penalized functions needed by the MCP and SCAD non-convex penalizations of Binomial, Poisson and Negative-Binomial related count responses regression. A case study of the colorectal cancer with six (6) covariates against sample size of five (5) was subjected to the non-convex penalized estimation of the three distributions. It was revealed that the non-convex penalization of Binomial regression via MCP and SCAD best explained four un-penalized covariates needed in determining whether surgical or therapy ideal for treating the turmoil.


Agarwal, A., Negahban, S. and Wainwright, M. J, “Fast global convergence of gradient methods for high-

dimensional statistical recovery” Annals of Statistics, vol. 40, no. 5, pp 2452–2482, 2012. doi:10.1214/12-AOS1032.

Chen, J. and Chen, Z., “Extended Bayesian information criteria for model selection with large model spaces”

Biometrika, vol. 95, no. 3, pp. 759–771, 2008.doi:10.1093/biomet/asn034.

Fan, J. and Li, R., “Variable selection via non-concave penalized likelihood and its oracle properties” Journal of the

American Statistical Association, vol. 96, no. 456, pp. 1348–1360, 2001. doi: 10.1198/016214501753882273.

Hirose, K and Yamamoto, M., “Penalized likelihood factor analysis via non-convex penalties”

th International Conference of the ERCIM WG on Computing & Statistics (ERCIM 2012), Conference

Centre, Oviedo, Spain, pp.1-33, December 2012.

Kim, Y., Kwon, S. and Choi, H, “Consistent model selection criteria on high dimensions” Journal of Machine

Learning Research, vol. 13, no. 1, pp. 1037-1057, 2012.

Liu, H.Y. Yao, T. and Li, R, “Global solutions to folded concave penalized non-convex learning” The Annals of

Statistics. Vol. 44, no. 2, pp. 629–659, 2016. doi:10.1214/15-AOS1380. © Institute of Mathematical Statistics.

Loh, P-L. and Wainwright, M. J, “Regularized M-estimators with non-convexity: Statistical and algorithmic theory

for local optima” Journal of Machine Learning Research, vol. 16, pp. 559–616, 2015.

Loh, P-L, “Local optima of non-convex regularized M-estimators” Electrical Engineering and Computer Sciences

University of California at Berkeley. Technical Report No. UCB/EECS-2013-54, 2013.

Nesterov, Y, “Gradient methods for minimizing composite functions” Mathematical Programming, vol.140, no. 1, pp.

–161, 2013. Doi:10.1007/s10107-012-0629-5.

Wang, Z., Han, H. and Zhang, T, “Optimal computational and statistical rates of convergence for sparse non-convex

learning problems” The Annals of Statistics, vol. 42, no. 6, pp. 2164–2201, 2014. doi: 10.1214/14-AOS1238.

Wang, L., Kim, Y. and Li, R, “Calibrating non-convex penalized regression in ultra-high dimension” Annals of

Statistics, vol. 41, no. 5, pp. 2505–2536, 2013. doi: 10.1214/13-AOS1159 © Institute of Mathematical Statistics.

Xiao, L., Zhang, T., “A proximal-gradient homotopy method for the sparse least-squares problem” SIAM Journal on

Optimization, vol. 23, no. 2, pp. 1062–1091, 2013.

Yang, N. and Han, L., “A general theory of hypothesis tests and confidence regions for sparse high dimensional

models” The Annals of Statistics, vol. 45, no. 1, pp. 158-195, 2017. doi:10.1214/16AOS1448.

Yu, Y. and Feng, Y., “APPLE: approximate path for penalized likelihood estimators” Journal of Statistical

Computing, vol. 24, pp. 803–819, 2014. doi: 10.1007/s11222-013-9403-7.

Zhang, C.H., “Nearly unbiased variable selection under minimax concave penalty” Annals of Statistics, vol. 38,

no. 2, pp. 894–942, 2010. doi: 10.1214/09-aos729.

Zhang, C.-H and Zhang, T., “A general theory of concave regularization for high dimensional sparse estimation

problems” Statistical Science, vol. 27, no. 4, pp. 576-593, 2012.

Zhang, K., Yin, F. and Xiong, F., “Comparisons of penalized least squares methods by simulations” 1-16, 2014.

arXiv:1405.1796v1 [stat. Co].

Zhong, L.W. and Kwok, J.F., “Gradient descent with proximal average for non-convex and composite

regularization, Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 1 – 7, 2014.




How to Cite

Olanrewaju, R. O. ., & Ojo, J. F. (2020). Non-Convex Penalized Estimation of Count Data Responses via Generalized Linear Model (GLM). Asian Journal of Fuzzy and Applied Mathematics, 8(3).