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Unlike the ordinary least-squares (OLS) estimator for the linear model, aridge regression linear model provides coefficient estimates via shrinkage,usually with improved mean-square and prediction error. This is true especiallywhen the observed design matrix is ill-conditioned or singular, either as aresult of highly-correlated covariates or the number of covariates exceedingthe sample size. This paper introduces novel and fast marginal maximumlikelihood (MML) algorithms for estimating the shrinkage parameter(s) for theBayesian ridge and power ridge regression models, and an automatic plug-in MMLestimator for the Bayesian generalized ridge regression model. With the aid ofthe singular value decomposition of the observed covariate design matrix, theseMML estimation methods are quite fast even for data sets where either thesample size (n) or the number of covariates (p) is very large, and even whenp>n. On several real data sets varying widely in terms of n and p, thecomputation times of the MML estimation methods for the three ridge models,respectively, are compared with the times of other methods for estimating theshrinkage parameter in ridge, LASSO and Elastic Net (EN) models, with the othermethods based on minimizing prediction error according to cross-validation orinformation criteria. Also, the ridge, LASSO, and EN models, and theirassociated estimation methods, are compared in terms of prediction accuracy.Furthermore, a simulation study compares the ridge models under MML estimation,against the LASSO and EN models, in terms of their ability to differentiatebetween truly-significant covariates (i.e., with non-zero slope coefficients)and truly-insignificant covariates (with zero coefficients).

Fast Marginal Likelihood Estimation of the Ridge Parameter(s) in Ridge Regression and Generalized Ridge Regression for Big Data