Quantile regression via vector generalized additive models

Abstract

One of the most popular methods for quantile regression is the LMS method of Cole and Green. The method naturally falls within a penalized likelihood framework, and consequently allows for considerable flexible because all three parameters may be modelled by cubic smoothing splines. The model is also very understandable: for a given value of the covariate, the LMS method applies a Box–Cox transformation to the response in order to transform it to standard normality; to obtain the quantiles, an inverse Box–Cox transformation is applied to the quantiles of the standard normal distribution. The purposes of this article are three-fold. Firstly, LMS quantile regression is presented within the framework of the class of vector generalized additive models. This confers a number of advantages such as a unifying theory and estimation process. Secondly, a new LMS method based on the Yeo–Johnson transformation is proposed, which has the advantage that the response is not restricted to be positive. Lastly, this paper describes a software implementation of three LMS quantile regression methods in the S language. This includes the LMS–Yeo–Johnson method, which is estimated efficiently by a new numerical integration scheme. The LMS–Yeo–Johnson method is illustrated by way of a large cross-sectional data set from a New Zealand working population. Copyright © 2004 John Wiley & Sons, Ltd.