Journal of Probability and Statistics

Volume 2019, Article ID 3493628, 11 pages

https://doi.org/10.1155/2019/3493628

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

^{1}School of Engineering, Computer & Mathematical Sciences, Auckland University of Technology, New Zealand^{2}Department of Statistics, University of Auckland, New Zealand

Correspondence should be addressed to V. F. Miranda-Soberanis; zn.ca.tua@adnarim.rotciv

Received 7 December 2018; Accepted 22 January 2019; Published 7 May 2019

Guest Editor: Rahim Alhamzawi

Copyright © 2019 V. F. Miranda-Soberanis and T. W. Yee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

##### 1.1. Background

Much of modern regression analysis for estimating conditional quantile functions may be viewed as starting from Koenker and Bassett [1], who offered a systematic strategy for examining how covariates influence the entire response distribution. The fundamental idea is based on the linear specification of the th quantile function and finding that solves the optimization problemfor independent and identically distributed (i.i.d.) observations from a family of linear quantile regression models, say . Equation (1) can be reformulated as a linear programming problem using the piecewise linear function for . More details can be found in Koenker [2].

In the spirit of quantile regression, the conditional distribution is usually unspecified, although it relies on normal–based asymptotic theory that is used for inference, whilst the assumption of homoskedasticity of the error terms is dropped. In this paper we use an alternative approach of conditional–quantile regression based on assuming a prespecified distribution for the response. Parametric quantile regression has some advantages over many nonparametric approaches, including overcoming the quantile crossing problem. Two examples are Noufaily and Jones [3] which is based on the generalized gamma distribution and generalized additive models for location, scale, and shape (GAMLSS; [4]). Further examples are the LMS-BCN method involving the standard normal distribution and a three–parameter Box–Cox transformation [5] and the classical method of quantile regression based on the asymmetric Laplace distribution (ALD).

Our approach uses the vector generalized linear and additive model (VGLM/VGAM; [6, 7]) framework. We develop new link functions, , for the quantile regression modelfor a vector of quantiles . Our methodology relies on the prespecification of the distribution . We will also show that the quantile crossing problem can be overcome by this modelling framework. Equations (2)–(3) state that the conditional distribution of the response at a given value of has a distribution involving a parameter and that the transformed quantile of the distribution becomes a linear predictor of the form (5). This can be achieved by defining link functions that connect (3) to (5). The reason for the linear predictors is that generalized linear modelling [8] is a very well-established method for regression modelling. GLMs are estimated by iteratively reweighted least squares (IRLS) and Fisher scoring, and this algorithm is also adopted by VGLMs and VGAMs.

The method presented in this paper differs from conventional quantile regression [1] in that we assume is known whereas the conventional case does not but use an empirical method instead to obtain the quantiles : the expectation of the check function results in the property which defines the -quantile ( is the cumulative distribution function (CDF) of ). In this paper we consider the s listed in Table 2.

##### 1.2. VGLMs and VGAMs

VGLMs/VGAMs provide the engine and overall modelling framework in this work—the VGAM R package described below fits over 150 models and distributions—therefore we only sketch the details here. VGLMs are defined in terms of linear predictors, , as any statistical model for which the conditional density of given a –dimensional vector of explanatory variables, has the formfor some known function , with , a matrix of unknown regression coefficients. Ordinarily, for an intercept.

In general, the of VGLMs may be applied directly to the parameters, , of any distribution, transformed if necessary, as the th linear predictorwhere is a VGLM–parameter link function, as in Table 1 (see [6] for further choices) and is the th element of . Prior to this work the were ‘raw’ parameters such as location, scale, and shape parameters; however, in this present work we define them to be quantiles or a very simple function of quantiles.