  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.
