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    High Dimensional Data Analysis: Integrating Submodels
    (Springer International Publishing Ag, 2017) Ahmed, Syed Ejaz; Yuzbasi, Bahadir
    We consider an efficient prediction in sparse high dimensional data. In high dimensional data settings where d >> n, many penalized regularization strategies are suggested for simultaneous variable selection and estimation. However, different strategies yield a different submodel with d(i) < n, where di represents the number of predictors included in ith submodel. Some procedures may select a submodel with a larger number of predictors than others. Due to the trade-off between model complexity and model prediction accuracy, the statistical inference of model selection becomes extremely important and challenging in high dimensional data analysis. For this reason we suggest shrinkage and pretest strategies to improve the prediction performance of two selected submodels. Such a pretest and shrinkage strategy is constructed by shrinking an overfitted model estimator in the direction of an underfitted model estimator. The numerical studies indicate that our post-selection pretest and shrinkage strategy improved the prediction performance of selected submodels.
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    LAD, LASSO and Related Strategies in Regression Models
    (Springer International Publishing Ag, 2020) Yuzbasi, Bahadir; Ahmed, Syed Ejaz; Arashi, Mohammad; Norouzirad, Mina
    In the context of linear regression models, it is well-known that the ordinary least squares estimator is very sensitive to outliers whereas the least absolute deviations (LAD) is an alternative method to estimate the known regression coefficients. Selecting significant variables is very important; however, by choosing these variables some information may be sacrificed. To prevent this, in our proposal, we can combine the full model estimates toward the candidate sub-model, resulting in improved estimators in risk sense. In this article, we consider shrinkage estimators in a sparse linear regression model and study their relative asymptotic properties. Advantages of the proposed estimators over the usual LAD estimator are demonstrated through a Monte Carlo simulation as well as a real data example.