This paper investigates risk bounds for quantile additive trend filtering, a method gaining increasing significance in the realms of additive trend filtering and quantile regression. We investigate the constrained version of quantile trend filtering within additive models, considering both fixed and growing input dimensions. In the fixed dimension case, we discover an error rate that mirrors the non-quantile minimax rate for additive trend filtering, featuring the main term $n^{-2r/(2r+1)}V^{2/(2r+1)}$, when the underlying quantile function is additive, with components whose $(r-1)$th derivatives are of bounded variation by $V$. In scenarios with a growing input dimension $d$, quantile additive trend filtering introduces a polynomial factor of $d^{(2r+2)/(2r+1)}$. This aligns with the non-quantile variant, featuring a linear factor $d$, particularly pronounced for larger $r$ values. Additionally, we propose a practical algorithm for implementing quantile trend filtering within additive models, using dimension-wise backfitting. We conduct experiments with evenly spaced data points or data that samples from a uniform distribution in the interval $[0,1]$, applying distinct component functions and introducing noise from normal and heavy-tailed distributions. Our findings confirm the estimator's convergence as $n$ increases and its superiority, particularly in heavy-tailed distribution scenarios. These results deepen our understanding of additive trend filtering models in quantile settings, offering valuable insights for practical applications and future research.

翻译：本文研究了分位数加性趋势滤波的风险界，该方法在加性趋势滤波和分位数回归领域中日益重要。我们探讨了加性模型中约束版本的分位数趋势滤波，同时考虑固定维度和增长维度两种情况。在固定维度情形下，当底层分位数函数具有加性结构、其各分量的$(r-1)$阶导数具有有界变差$V$时，我们发现其误差率与非分位数加性趋势滤波的极小极大速率一致，主要项为$n^{-2r/(2r+1)}V^{2/(2r+1)}$。在输入维度$d$增长的情形下，分位数加性趋势滤波引入了多项式因子$d^{(2r+2)/(2r+1)}$。这与非分位数变体一致，后者在线性因子$d$上表现显著，尤其当$r$取值较大时。此外，我们提出了一种实用算法，利用逐维回拟合方法实现加性模型中的分位数趋势滤波。我们针对均匀分布间隔数据点或从区间$[0,1]$均匀分布中采样的数据进行了实验，应用不同分量函数并引入正态分布和重尾分布的噪声。实验结果验证了该估计量随$n$增大而收敛的特性，以及其在重尾分布场景下的优越性。这些结果深化了我们在分位数设定下对加性趋势滤波模型的理解，为实际应用和未来研究提供了宝贵见解。