A NONPARAMETRIC BOOTSTRAP LIKELIHOOD RATIO TEST FOR QUANTILE REGRESSION

Abstract

Likelihood ratio based confidence intervals often better accommodate asymmetric un- certainty about parameter estimates compared to other likelihood-based approaches. In practice, likelihood ratio tests (LRT) are mostly determined by chi-square thresh- olds or parametric bootstrapping thresholds, which give valid coverage probabilities under the assumption that parametric models are correct. But, in settings where likelihood estimation is robust to model misspecification, it is often the case that the likelihood theory leading to hypothesis tests and confidence intervals breaks down. In this work, a nonparametric bootstrapping approach to LRT was developed to determine the critical value for misspecified data in the context of quantile regression models. The performance of the nonparametric LRT is compared with commonly used tests via simulated and real data. Examples of asymmetric Laplace distribution and quantile regression will be focused upon in the comparison. In addition a fast normal approximation of percentile method is derived in this thesis. This thesis will show that compared to the Wald test, chi-square LRT, percentile method, and percentile-t method, the nonparametric bootstrapping likelihood ra- tio test often provides better confidence intervals. Finally, methods are illustrated through a real data example.