  When the unknown regression function of a single variable is known to have
derivatives up to the $(\gamma+1)$th order bounded in absolute values by a
common constant everywhere or a.e. (i.e., $(\gamma+1)$th degree of smoothness),
the minimax optimal rate of the mean integrated squared error (MISE) is stated
as $\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ in the literature.
This paper shows that: (i) if $n\leq\left(\gamma+1\right)^{2\gamma+3}$, the
minimax optimal MISE rate is $\frac{\log n}{n\log(\log n)}$ and the optimal
degree of smoothness to exploit is roughly $\max\left\{ \left\lfloor \frac{\log
n}{2\log\left(\log n\right)}\right\rfloor ,\,1\right\} $; (ii) if
$n>\left(\gamma+1\right)^{2\gamma+3}$, the minimax optimal MISE rate is
$\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ and the optimal degree
of smoothness to exploit is $\gamma+1$. The fundamental contribution of this
paper is a set of metric entropy bounds we develop for smooth function classes.
Some of our bounds are original, and some of them improve and/or generalize the
ones in the literature (e.g., Kolmogorov and Tikhomirov, 1959). Our metric
entropy bounds allow us to show phase transitions in the minimax optimal MISE
rates associated with some commonly seen smoothness classes as well as
non-standard smoothness classes, and can also be of independent interest
outside the nonparametric regression problems.
