New Link Functions for Distribution–Specific Quantile Regression Based on Vector Generalized Linear and Additive Models

Abstract

In the usual quantile regression setting, the distribution of the response given the explanatory variables is unspecified. In this work, the distribution is specified and we introduce new link functions to directly model specified quantiles of seven 1–parameter continuous distributions. Using the vector generalized linear and additive model (VGLM/VGAM) framework, we transform certain prespecified quantiles to become linear or additive predictors. Our parametric quantile regression approach adopts VGLMs/VGAMs because they can handle multiple linear predictors and encompass many distributions beyond the exponential family. Coupled with the ability to fit smoothers, the underlying strong assumption of the distribution can be relaxed so as to offer a semiparametric–type analysis. By allowing multiple linear and additive predictors simultaneously, the quantile crossing problem can be avoided by enforcing parallelism constraint matrices. This article gives details of a software implementation called the VGAMextra package for R. Both the data and recently developed software used in this paper are freely downloadable from the internet.