Shrinkage Parameters for Each Explanatory Variable Found Via Particle Swarm Optimization in Ridge Regression

Ridge regression method is an improved method when the assumptions of independence of the explanatory variables cannot be achieved, which is also called multicollinearity problem, in regression analysis. One of the way to eliminate the multicollinearity problem is to ignore the unbiased property of .Ridge regression estimates the regression coefficients biased in order to decrease the variance of the regression coefficients. One of the most important problems in ridge regression is to decide what the shrinkage parameter (k) value will be. This k value was found to be a single value in almost all these studies in the literature. In this study, different from those studies, we found different k values corresponding to each diagonal elements of variance-covariance matrix of  instead of a single value of k by using a new algorithm based on particle swarm optimization. To evaluate the performance of our proposed method, the proposed method is firstly applied to real-life data sets and compared with some other studies suggested in the ridge regression literature. Finally, two different simulation studies are performed and the performance of the proposed method with different conditions is evaluated by considering other studies suggested in the ridge regression literature..