Additive Models for Quantile Regression: Some New Methods for R

This brief report describes some recent developments of the R quantreg package to incorporate methods for additive models. The methods are illustrated with an application to modeling childhood malnutrition in India. Models with additive nonparametric effects offer a valuable dimension reduction device throughout applied statistics. In this paper we describe some recent developments of additive models for quantile regression. These methods employ the total variation smoothing penalties introduced in Koenker, Ng, and Portnoy (1994) for univariate components and Koenker and Mizera (2004) for bivariate components. We focus on selection of smoothing parameters including lasso-type selection of parametric components, and on post selection inference methods. Additive models have received considerable attention since their introduction by Hastie and Tibshirani (1986, 1990). They provide a pragmatic approach to nonparametric regression modeling; by restricting nonparametric components to be composed of low-dimensional additive pieces we can circumvent some of the worst aspects of the notorious curse of dimensionality. It should be emphasized that we use the word “circumvent’ advisedly, in full recognition that we have only swept difficulties under the rug by the assumption of additivity. When conditions for additivity are violated there will obviously be a price to pay. 1. Additive Models for Quantile Regression Our approach to additive models for quantile regression and especially our implementation of methods in R is heavily influenced by Wood (2006, 2009) . In some fundamental respects the approaches are quite distinct: Gaussian likelihood is replaced by (Laplacean) quantile fidelity, squared L2 norms as measures of the roughness of fitted functions are replaced by corresponding L1 norms measuring total variation, and truncated basis expansions are supplanted by sparse algebra as a computational expedient. But in other respects the structure of the models is quite similar. We will consider models for conditional quantiles of the general form: Version: September 5, 2009. This research was partially supported by NSF grant SES-08-50060. I would like to express my appreciation to Ying Li for excellent research assistance. All of the methods described below have been implemented in version 4.42 of the quantreg package for R, Koenker (2009).