Superquantile Regression: Theory, Algorithms, and Applications.

摘要

We present a novel regression framework centered on a coherent and averse measure of risk, the superquantile risk (also called conditional value-at-risk), which yields more conservatively fitted curves than classical least squares and quantile regressions. In contrast to other generalized regression techniques that approximate conditional superquantiles by various combinations of conditional quantiles, we directly and in perfect analog to classical regression obtain superquantile regression functions as optimal solutions of certain error minimization problems. We show the existence and possible uniqueness of regression functions, discuss the stability of regression functions under perturbations and approximation of the underlying data, and propose an extension of the coefficient of determination R-squared and Cook's distance for assessing the goodness of fit for both quantile and superquantile regression models. We present two classes of computational methods for solving the superquantile regression problem, compare both methods' complexity, and illustrate the methodology in eight numerical examples in the areas of military applications, concerning mission employment of U.S. Navy helicopter pilots and Portuguese Navy submariners, reliability engineering, uncertainty quantification, and financial risk management.