Simultaneous multiple non-crossing quantile regression estimation using kernel constraints

Abstract

Quantile regression (QR) is a very useful statistical tool for learning the relationship between the response variable and covariates. For many applications, one often needs to estimate multiple conditional quantile functions of the response variable given covariates. Although one can estimate multiple quantiles separately, it is of great interest to estimate them simultaneously. One advantage of simultaneous estimation is that multiple quantiles can share strength among them to gain better estimation accuracy than individually estimated quantile functions. Another important advantage of joint estimation is the feasibility of incorporating simultaneous non-crossing constraints of QR functions. In this paper, we propose a new kernel-based multiple QR estimation technique, namely simultaneous non-crossing quantile regression (SNQR). We use kernel representations for QR functions and apply constraints on the kernel coefficients to avoid crossing. Both unregularised and regularised SNQR techniques are considered. Asymptotic properties such as asymptotic normality of linear SNQR and oracle properties of the sparse linear SNQR are developed. Our numerical results demonstrate the competitive performance of our SNQR over the original individual QR estimation.

Keywords:

• asymptotic normality
• kernel
• multiple quantile regression
• non-crossing
• oracle property
• regularisation
• variable selection

Acknowledgements

Liu's research was supported in part by NSF grant DMS-0747575 and NIH grant 1R01CA149569-01. Wu's research was supported in part by NSF grant DMS-0905561 and NIH grant 1R01CA149569-01. The authors are indebted to the editor, the associate editor and two referees, whose helpful comments and suggestions led to a much improved presentation.