On the Efficiency of the newly Proposed Convex Olanrewaju-Olanrewaju Lo-oλγ(|θ|) Penalized Regression-Type Estimator via GLMs Technique.

In this article, we proposed a novel convex penalized regression-type estimator, termed Olanrewaju-Olanrewaju penalized regression-type estimator, denoted by  Lo-oλγ(|θ|) for ultra and high-dimensional data consideration whenever the number of covariates (p) is strictly greater than the sample size(n⁡). The newly proposed penalized estimator was formulated via the unique combination of arctangent and hyperbolic functions. It was noted that the newly proposed estimator could efficiently work under the constraint p<n formulated for, and as well as for the conventional regression methods of n>p. The proposed convex penalized regression-type estimator was formulated via three symmetric distributional noises of Gaussian, Laplace, and Normal Inverse Gaussian (NIG), considered being the three certified symmetric distributions that belong to the exponential family of distributions. The regularity axioms and oracle properties of the newly proposed penalized estimator were worked-out, as well as the parameter estimation for the penalized regression-type estimator for the three symmetric noises via canonical expressible form of Generalized Linear Models (GLMs) and first-order Karush-Kuhn-Tucker (KKT) condition. The Lo-oλγ(|θ|)  regression-type estimator was subjected to simulation studies of well-generated symmetric observations. The US Longley macroeconomic dataset with the constraint p<n was fitted to Lo-oλγ(|θ|) regression-type estimator under the umbrella of three symmetric distributional noises. In conclusion, Gaussian noise for Lo-oλγ(|θ|)  penalized regression-type estimator under the umbrella of p<n for the simulation and real life analyzes was considered effectively efficient because of its most reduced Mean Square Error (mse) measured error-type in comparison to Gaussian, Laplace, and NIG noises for LASSO and Atan penalties; and as well for Laplace, and NIG noises for Lo-oλγ(|θ|).