Applications of quantile regression to estimation and detection of some tail characteristics

The statistical inference based on the ordinary least squares regression is sub-optimal when the distributions are skewed or when the quantity of interest is the upper or lower tail of the distributions. For example, the changes in Total Sharp Scores (TSS), the primary measurements of the treatment effects on prevention of structural damage for rheumatoid arthritis, are nearly identical for most therapies for nearly 75% of the patient population, but the difference lies in the most challenging 25% of the patient population where a less effective treatment loses its efficacy, resulting in a heavy right tail in its distribution.In the first part of the dissertation, we develop the Expected Shortfall (ES), the Covariate-adjusted Expected Shortfall (COVES), and the Generalized Covariate-adjusted Expected Shortfall (q.COVES) tests under the framework of quantile regression. Those tests focus specifically on one tail of the outcome distributions. The ES test applies to two-sample comparisons. The COVES test adjusts for covariates, and is shown to be valid for i.i.d (independent and identically distributed) error models or when the covariates have the same means across treatments. The q.COVES test generalizes the COVES test to more general models. We show the proposed tests can achieve a substantial sample size reduction over the conventional tests on mean effects.The second part of the dissertation focuses on a popular measure of risk used by financial institutions, Value at Risk (VaR), defined as a quantile of the loss distribution of a portfolio within a given time period and a confidence level. Accurate VaR estimation can help financial institutions maintain appropriate capital levels to cover the risk from the corresponding portfolio. We use an MCMC strategy along with a block algorithm to perform Bayesian inference on the Conditional Autoregressive Value at Risk (CAViaR) models proposed by Engle and Manganelli (2004) based on quantile regression. Using the S&P 500 index as an example, we show that the proposed Bayesian approach adds value to the original estimation method of Engle and Manganelli in terms of both estimation and prediction.