Abstract: Understanding the heterogeneous covariate-response relationship is central to modern data analysis. Beyond the usual descriptors such as the mean and variance, quantile and superquantile (also known as the expected shortfall or conditional value-at-risk) regression can capture the differential covariate effects on the upper or lower tails of the response distribution. This dissertation studies some fundamental aspects of the statistical inference of quantile and super quantile regression.

In the first part of the dissertation, we propose a novel approach to superquantile regression with a critical modification of an optimization formulation in the recent literature. Most existing approaches for superquantile regression rely explicitly on the modeling of the conditional quantile function. In this dissertation, we offer new insights into an optimization formulation for the superquantile, based on which we provide and validate a direct approach to superquantile regression estimation without relying on additional quantile regression modeling. Operationally, the approach can be well approximated by fitting a linear quantile regression to an array of pre-estimated conditional superquantile processes. This approach achieves implicit weighting of the data, which is found to be automatically adaptive to data heterogeneity in a variety of scenarios. With certain initial estimators based on binning of the covariate space, we show the proposed superquantile regression estimator is consistent and asymptotically normal. Via theoretical and numerical comparisons, we show that the proposed approach has competitive, and often superior, performance relative to other common approaches in the literature.

In the second part of the dissertation, we study pseudo-Bayesian inference for possibly sparse quantile regression models. We find that by coupling the asymmetric Laplace working likelihood with appropriate shrinkage priors, we can deliver pseudo-Bayesian inference that adapts automatically to the possible sparsity in quantile regression analysis. After a suitable adjustment on the posterior variance, the proposed method provides asymptotically valid inference under heterogeneity. Furthermore, the proposed approach leads to oracle asymptotic efficiency for the active (nonzero) quantile regression coefficients and super-efficiency for the non-active ones. We also discuss the theoretical extension when the covariate dimension increases with the sample size at a controlled rate. By avoiding the need to pursue dichotomous variable selection as well as nuisance parameter estimation, the Bayesian computational framework demonstrates desirable inferential stability.