It is well known that many penalized regression problems can be interpreted as estimating unknown regression coefficients having assumed a specific statistical model. This includes the lasso when tuning parameters are estimated from the marginal likelihood of the data, the Bayesian lasso, Gaussian random effects models, ridge regression, etc. In the first part, we consider estimating a mean matrix from a single noisy realization. We assume possibly sparse elementwise effects and use a lasso penalty. In this setting the elementwise effects are unreplicated and selecting the tuning parameters using cross validation is not possible without imposing additional assumptions on the mean matrix. Instead, we use the model interpretation to construct moment-based Empirical Bayes estimators of the tuning parameters that are easy and fast to compute, even when the data is high dimensional. When studying the robustness of this method to misspecification of the distribution of elementwise effects, we observe that misspecification causes poor performance of the tuning parameter estimators in some settings. Accordingly in the second part, we consider the standard lasso penalized regression model and introduce a test for detecting misspecification of the penalty. We show that this test performs well when the number of observations is large regardless of the number of penalized regression coefficients, and that the test statistic can be used to adaptively specify a more appropriate penalized regression model after rejection. Lastly, we consider a different kind of misspecification of penalized regression models: misspecification of the correlation structure of penalized regression coefficients. This last topic is the focus of our future work. Most models assume penalized regression coefficients are independent. Because that is unlikely in many settings, we describe a new penalized regression model that can accommodate correlated regression parameters.
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