### Web: http://arxiv.org/abs/2112.03626

June 17, 2022, 1:11 a.m. | Ying Zhu

When the regression function belongs to the smooth classes consisting of
univariate functions with derivatives up to the $(\gamma+1)$th order bounded in
absolute values by a common constant everywhere or a.e., it is generally viewed
that exploiting higher degree smoothness assumption helps reduce the estimation
error. This paper shows that the minimax optimal mean integrated squared error
(MISE) rate increases in $\gamma$ when the sample size $n$ is small relative to
$\left(\gamma+1\right)^{2\gamma+3}$ (e.g.,
$\left(\gamma+1\right)^{2\gamma+3}=262144$ when $\gamma=3$), and decreases in
$\gamma$ …

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