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.

中文翻译：

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使用核约束的同时多重非交叉分位数回归估计

分位数回归 (QR) 是一种非常有用的统计工具，用于学习响应变量和协变量之间的关系。对于许多应用程序，通常需要估计给定协变量的响应变量的多个条件分位数函数。虽然可以分别估计多个分位数，但同时估计它们很有趣。同时估计的一个优点是多个分位数可以共享它们之间的强度，以获得比单独估计的分位数函数更好的估计精度。联合估计的另一个重要优势是合并 QR 函数的同时非交叉约束的可行性。在本文中，我们提出了一种新的基于内核的多重 QR 估计技术，即同时非交叉分位数回归（SNQR）。我们对 QR 函数使用核表示，并对核系数应用约束以避免交叉。考虑了非正则化和正则化 SNQR 技术。开发了渐近特性，例如线性 SNQR 的渐近正态性和稀疏线性 SNQR 的预言机特性。我们的数值结果证明了我们的 SNQR 相对于原始个体 QR 估计的竞争性能。