| Model and variable selection is an important and integral component in linear regression analysis. Traditional approaches focus on developing suitable objective functions and, as a result, reducing the selection to an optimization problem. There are renewed interests in recent years, mainly on developing simultaneous parameter estimation and variable selection, especially for problems where number of explanatory variables is large. A notable contribution in this direction is due to Tibshirani who developed the Least Absolute Shrinkage and Selection Operator (LASSO).; This thesis addresses the problem of variable selection in the context of L1 regression. It introduces a LASSO-type penalty to the L1 criterion function. The resulting procedure can be implemented by the efficient linear programming algorithm. Large sample properties are derived via modern empirical process theory and certain novel inequalities. Estimation of standard errors is obtained by applying a computationally intensive resampling scheme that avoids numerical differentiation and density estimation. An adaptive two-stage approach is proposed to achieve the so-called oracle property-simultaneously achieving consistent variable selection and best parameter estimation. A parallel approach is developed for the rank estimation in linear regression. This is important due to its link to the accelerated failure time (AFT) model in survival analysis where the response variable is often subject to right censorship. The new method handles the right censoring in a natural way. Desirable large sample properties, including consistency, asymptotic normality, and the oracle property are derived for the proposed method.; Extensive simulations are conducted to assess finite-sample performances of the proposed methods. Comparisons with existing approaches are made through simulation studies. Illustrations with real examples are provided. |
