The Least absolute deviation combined with the Least absolute shrinkage andselection operator (LAD-LASSO) estimator can do regression shrinkage and selectionand is also resistant to outliers or heavy-tailed errors which is proposed in Wang etal. (2007). Generalized likelihood ratio (GLR) test motivated by the likelihood principle, which does not require knowing the underlying distribution family andalso shares the Wilks property, has wide applications and nice interpretations [cf.Fan et al. (2001) and Fan and Jiang (2005)]. In this dissertation, we propose aGLR test based on LAD-LASSO estimators in order to combine their advantagestogether. We obtain the asymptotic distributions of the test statistic by applying theBahadur representation to the LAD-LASSO estimators. Furthermore, we show thatthe test has oracle property and can detect alternatives nearing the null hypothesis ata maximum rate of root-n. Simulations are conducted to compare test statistics underdifferent procedures for a variety of error distributions including standard normal, t3and mixed normal. A real data example is used to illustrate the performance of thetesting approach.
