Model selection, inference strategies and their applications

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Ascertaining the appropriate statistical model-estimator for use in representing the data sampling process is an interesting and challenging problem in statistical research. In the main part of this proposal, we consider model selection and post model parameter estimation strategies in a host of scenarios and applications. Recent literature on variable selection focuses on using the data from an experiment to find a candidate subspace that represents a sparsity pattern in the predictor space. In the next round of experiments, researchers may consider this information and use either the full model or the candidate submodel. The strategy in this project is inspired by Stein's result that, in dimensions greater than two, efficient estimates can be obtained by shrinking full model estimates in the direction of submodel estimates. In some studies, many covariates are collected and included in the initial model. Because this may increase the uncertainty of the results, variable selection will be a crucial part of statistical analysis. Parsimony and reliability of predictors are desirable characteristics of statistical models. One possible source of prior information consists of knowing which of the predictor variables are of main interest and which variables are nuisance variables such as lab or age (candidate confounders) that may not affect the analysis of the association between the response and the main predictors. Another source of prior information is knowledge of results from previous experiments that search for sparsity patterns. This knowledge can be used to propose candidate subspaces. However, shrinking the full model estimator in the direction of the subspace leads to more efficient estimators when the shrinkage is adaptive and based on the estimated distance between the subspace and the full space estimators. We will establish risk properties of the shrinkage estimators via asymptotic distributional risk, a novel approach, and Monte Carlo experiments. It is expected that the proposed research will provide a unified strategy to researchers and practitioners for inference after variable selection and in assessing the predictive ability of the model at hand.