A Comparison of Bootstrap Methods for Adaptive LASSO + Partial Ridge to Construct Confidence Intervals for Parameters in High-Dimensional Sparse Linear Models

Abstract

This research is aimed to propose a method, called bootstrap adaptive lasso + partial ridge (ALPR), to construct confidence intervals of regression coefficients in high – dimensional data and compare its performance with bootstrap lasso + partial ridge (LPR). The ALPR is a two-stage estimator. The adaptive lasso is used to select variables and the partial ridge is used to refit the coefficients. Here we perform two techniques of bootstrap which are residual bootstrap and paired bootstrap. We also consider two cases of coefficients which are weak sparsity and hard sparsity where weak sparsity and hard sparsity refer to the case that majority of coefficients have value closed to zero and equal to zero respectively. Simulation studies in 8 cases of high – dimensional covariates that are generated from multivariate normal distribution with different types of covariance matrix. Mean interval lengths and coverage probabilities are used to measure and compare performance of bootstrap methods. Our simulation studies show that the residual bootstrap adaptive lasso + partial ridge provides lowest mean interval lengths for most cases. However, it is not obvious that which bootstrap method is the best in terms of providing highest coverage probabilities. We also apply each bootstrap method with the real data, colon cancer microarray data set. The results show that the residual bootstrap adaptive lasso + partial ridge and the paired bootstrap adaptive lasso + partial ridge are the best method in terms of mean interval lengths and coverage probabilities, respectively.