论文标题
用深层神经网络估算非交叉的分位回归过程
Estimation of Non-Crossing Quantile Regression Process with Deep ReQU Neural Networks
论文作者
论文摘要
我们提出了一种惩罚的非参数方法，用于使用整流器二次单元（reque）在不可分割的模型中估算分位数回归过程（QRP）激活了深层神经网络，并引入了新的惩罚函数，以实施对瓦解回归曲线的非交叉。我们为估计的QRP建立了非反应过量的风险界限，并在轻度平滑度和规律性条件下得出了估计的QRP的平均综合平方误差。为了建立这些非反应风险和估计误差界，我们还开发了一个新的错误，用于使用$ s> 0 $的$ C^s $平滑功能及其衍生物使用所需的激活神经网络。这是必需网络的新近似结果，并且具有独立的兴趣，并且可能在其他问题中有用。我们的数值实验表明，所提出的方法具有竞争性或胜过两种现有方法，包括使用复制核和随机森林进行非参数分位数回归的方法。
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.
