A New Ridge-Type Estimator for the Gamma Regression Model.

Adewale F Lukman,B M Golam Kibria,Issam Dawoud,Zakariya Y Algamal,Benedicta Aladeitan,Francisco Ayuga

doi:10.1155/2021/5545356

Abstract

The known linear regression model (LRM) is used mostly for modelling the QSAR relationship between the response variable (biological activity) and one or more physiochemical or structural properties which serve as the explanatory variables mainly when the distribution of the response variable is normal. The gamma regression model is employed often for a skewed dependent variable. The parameters in both models are estimated using the maximum likelihood estimator (MLE). However, the MLE becomes unstable in the presence of multicollinearity for both models. In this study, we propose a new estimator and suggest some biasing parameters to estimate the regression parameter for the gamma regression model when there is multicollinearity. A simulation study and a real-life application were performed for evaluating the estimators' performance via the mean squared error criterion. The results from simulation and the real-life application revealed that the proposed gamma estimator produced lower MSE values than other considered estimators.

Highlights

Scientifica e main objective portrayed in this article is to extend the new ridge-type estimator of Kibria and Lukman [17] to the gamma regression model (GRM). e article organization is as follows: in Section 1, we proposed the new ridge-type gamma estimator, and we derived its properties
Where β∗i would be any of the following estimators (MLE, gamma ridge estimator (GRE), gamma Liu estimator (GLE), and GLK). e smaller the mean square error value is, the better the estimator is. e biasing parameters for GRE and GLE are obtained as follows: p k􏽢 min⎛⎜⎜⎝β􏽢′M2Lφ􏽢E(j)⎞⎟⎟⎠j 1, d􏽢
Since the performance of the proposed estimator gamma KL estimator (GKL) depends on its biasing parameter, we examined two different biasing parameters for GKL estimator and observed that the GKL estimator performs best with the biasing parameter, k􏽢2. e simulation result further supports the theoretical results that the performance of GKL estimator is the best. e performance of the GRE and GLE is better than that of the maximum likelihood estimator (MLE)

Summary

The Statistical Methodology

Consider the response variable yi which follows the known gamma distribution with the parameter of the nonnegative shape aand the parameter of the nonnegative scale b with probability density function:. Where cj is considered as an jth eigenvalue of the given matrix D X′ W􏽢 X and the notation X′is the transpose of X. where Dk (Ι + kD− 1) and k is the biasing parameter. E MMSE and MSE of GRE are given by. Where α P′β such that P is the matrix of eigenvectors of D. e gamma Liu estimator (GLE) is given by β􏽢GLE Fdβ􏽢MLE, 0 < d < 1,. E MMSE and MSE of GLE are given by. E bias and covariance matrix form of GKL estimator are gotten respectively as: Bias􏼐β􏽢GKL􏼑 􏼐D−k1Rk − Ι􏼑β,. Some needed lemmas are stated as follows for comparing the estimators in theoretical.

Comparison of GKL and GRE

Comparison of GKL and GLE

Simulation Design

Real-Life Data