Hamid Karamikabir

This thesis investigates Bridge regression as a flexible method within the family of penalized regression techniques, which, by employing the ‎L‎_\alpha penalty, offers a generalization of both Ridge and Lasso methods. The main challenge in applying such methods is the selection of tuning parameters and the computational complexity of iterative solution approaches such as MCMC sampling and the EM algorithm. This thesis addresses these limitations by presenting a non-iterative, closed-form-based approach. The core idea relies on using a normal scale mixture representation for the power exponential distribution (the Bridge prior), which enables the analytical computation of posterior moments. The proposed SURE-Bridge method utilizes the latent normal representation to derive Stein's Unbiased Risk Estimate (SURE) for Bridge regression, allowing the tuning parameter to be selected via one-dimensional numerical optimization. This approach entirely avoids expensive iterative procedures and reduces computations to simple, vectorizable Monte Carlo simulations. Simulation results and an application to spectroscopic data demonstrate that the proposed SURE-Bridge method achieves competitive statistical performance compared to fully Bayesian methods and cross-validation, while being significantly faster in terms of computational time and exhibiting favorable scalability as data dimensionality increases.