$L_q$-type Penalty Estimation Under the Linear Restriction

This study is concerned with the Bridge Regression, which is a special family of penalized regressions of a penalty function $\sum_{j=1}^{p}|\beta_j|^q$ with $q>0$, in a linear model with linear restrictions. This estimator helps to estimate when it is known prior information regarding data that may come from both low dimensional case or high dimensional case. Using local quadratic approximation, the penalty term can be approximated around a local initial values vector and the restricted bridge estimation has written a closed-form which can be solved when $q>0$. Special cases of our suggested estimation are restricted LASSO ($q=1$) and restricted RIDGE ($q=2$) and restricted Elastic Net ($1< q < 2$) estimators. It is given some theoretical property of the restricted bridge estimator as well as computational details. A Monte Carlo simulation study is conducted based on different prior pieces of information and compared its performance with some competitive penalty estimators as well as ORACLE. Also, it is considered four real-world data examples. The numerical results show that the suggested estimation outstandingly performs when the preliminary information is correct or close to accuracy.