HDBRR: A Statistical Package for High Dimensional Ridge Regression without MCMC

Ridge regression is a useful tool to deal with collinerity in the homoscedastic linear regression model, which provide biased estimators of the regression parameters with lower variance than the least square estimators. Evenmore, when the number of predictors (p) is much larger than the number of observations (n), ridge regression give us unique least square estimators by restringing the parametric space to the neighborhood of the origin. From the Bayesian point of view ridge regression results of assigning a Gaussian prior on the regression parameters and assuming they are conditionally independent. However, from both classical and Bayesian approaches the estimation of parameters is a highly demanding computational task, in the first one being an optimization problem and in the second one a high dimensional integration problem usually faced up through Markov Chain Monte Carlo (MCMC). The main drawback of MCMC is the practical impossibility of checking convergence to the posterior distribution, which is commonly very slow due to the large number of regression parameters. Here we propose a computational algorithm to obtain posterior estimates of regression parameters, variance components and predictions for the conventional ridge Regression model, based on a reparameterization of the model which allows us to obtain the marginal posterior means and variances by integrating out a nuisance parameter whose marginal posterior is defined on the open interval (0, 1).