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L0 penalized likelihood procedures like Mallows' Cp, AIC, and BIC directly penalize for the number of variables included in a regression model. This is a straightforward approach to the problem of overfitting, and these methods are now part of every statistician's repertoire. However, these procedures have been shown to sometimes result in unstable parameter estimates as a result on the L0 penalty's discontinuity at zero. One proposed alternative, seamless-L0 (SELO), utilizes a continuous penalty function that mimics L0 and allows for stable estimates. Like other similar methods (e.g. LASSO and SCAD), SELO produces sparse solutions because the penalty function is non-differentiable at the origin. Because these penalized likelihoods are singular (non-differentiable) at zero, there is no closed-form solution for the extremum of the objective function. We propose a continuous and everywhere-differentiable penalty function that can have arbitrarily steep slope in a neighborhood near zero, thus mimicking the L0 penalty, but allowing for a nearly closed-form solution for the beta-hat vector. Because our function is not singular at zero, beta-hat will have no zero-valued components, although some will have been shrunk arbitrarily close thereto. We employ a BIC-selected tuning parameter used in the shrinkage step to perform zero-thresholding as well. We call the resulting vector of coefficients the ShrinkSet estimator. It is comparable to SELO in terms of model performance (selecting the truly nonzero coefficients, overall MSE, etc.), but we believe it to be more intuitive and simpler to compute. We provide strong evidence that the estimator enjoys favorable asymptotic properties, including the oracle property.

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For a linear regression, the traditional technique deals with a case where the number of observations n more than the number of predictor variables p (n > p). In the case n < p, the classical method fails to estimate the coefficients. A solution of the problem is the case of correlated predictors is provided in this thesis. A new regularization and variable selection is proposed under the name of Sparse Ridge Fusion (SRF). In the case of highly correlated predictor, the simulated examples and a real data show that the SRF always outperforms the lasso, eleastic net, and the S-Lasso, and the results show that the SRF selects more predictor variables than the sample size n while the maximum selected variables by lasso is n size.

This dissertation develops a comprehensive statistical learning framework that is robust to (distributional) perturbations in the data using Distributionally Robust Optimization (DRO) under the Wasserstein metric. The learning problems that are studied include: (i) Distributionally Robust Linear Regression (DRLR), which estimates a robustified linear regression plane by minimizing the worst-case expected absolute loss over a probabilistic ambiguity set characterized by the Wasserstein metric; (ii) Groupwise Wasserstein Grouped LASSO (GWGL), which aims at inducing sparsity at a group level when there exists a predefined grouping structure for the predictors, through defining a specially structured Wasserstein metric for DRO; (iii) Optimal decision making using DRLR informed K-Nearest Neighbors (K-NN) estimation, which selects among a set of actions the optimal one through predicting the outcome under each action using K-NN with a distance metric weighted by the DRLR solution; and (iv) Distributionally Robust Multivariate Learning, which solves a DRO problem with a multi-dimensional response/label vector, as in Multivariate Linear Regression (MLR) and Multiclass Logistic Regression (MLG), generalizing the univariate response model addressed in DRLR. A tractable DRO relaxation for each problem is being derived, establishing a connection between robustness and regularization, and obtaining upper bounds on the prediction and estimation errors of the solution. The accuracy and robustness of the estimator is verified through a series of synthetic and real data experiments. The experiments with real data are all associated with various health informatics applications, an application area which motivated the work in this dissertation. In addition to estimation (regression and classification), this dissertation also considers outlier detection applications.

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