On Some Aspects of Model Selection Variability

In this thesis, we investigate the data analytic approach to integrate the model selection uncertainty into the statistical inferences of high dimensional estimators. Two closed-form formulae of covariance matrices are derived for high dimensional bagging estimators, one for the nonparametric bootstrapping and the other for the parametric bootstrapping. Two simulation studies are completed in detail for demonstrating the validity of the new formulae. Several model selection methods --- the hypothesis testing, the Mallows' $Cp$, AIC, BIC and LASSO --- are compared in terms of the effects on the accuracy of bagging estimators in the context of multivariate linear regression. The confidence region and its coverage probability are also estimated for the bagging estimators with those model selection methods.