We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating $C^s$ smooth functions with $s >0$ and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests, for nonparametric quantile regression.

翻译：我们提出一种惩罚性非对称方法,用一种不可分离模型来估计四分位回归过程(QRP),使用矩形二次单位(ReQU)激活深神经网络,并引入一种新的惩罚功能,以强制四分位回归曲线的不交叉。我们为估计的QRP建立了非非非被动过度风险边框,并在温和和正常的条件下为估计的QRP得出平均综合方差。为了确定这些非被动风险和估计误差界限,我们还开发了一个新的错误,用于约合美元 >0美元的光滑功能及其衍生物,使用ReQU激活神经网络。这是REQU网络的新近似结果,具有独立的兴趣,并且可能对其他问题有用。我们的数字实验表明,拟议的方法与两种现有方法具有竞争力,或超出两种现有方法,包括使用再生内核和随机森林的方法,用于非对定量二次回归。