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.

翻译：当单变量未知回归函数已知具有直至$(\gamma+1)$阶的有界导数，且其绝对值处处或几乎处处被一个公共常数所控制（即$(\gamma+1)$阶光滑度）时，文献中给出的均方积分误差（MISE）的极小极大最优速率为$\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$。本文表明：（i）若$n\leq\left(\gamma+1\right)^{2\gamma+3}$，则极小极大最优MISE速率为$\frac{\log n}{n\log(\log n)}$，且需利用的最优光滑度阶数近似为$\max\left\{ \left\lfloor \frac{\log n}{2\log\left(\log n\right)}\right\rfloor ,\,1\right\}$；（ii）若$n>\left(\gamma+1\right)^{2\gamma+3}$，则极小极大最优MISE速率为$\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$，且需利用的最优光滑度阶数为$\gamma+1$。本文的根本贡献在于为光滑函数类建立了一套度量熵界。其中部分熵界为原创性结果，另一些则改进和/或推广了文献中的已有结论（例如Kolmogorov与Tikhomirov, 1959）。我们的度量熵界不仅能揭示若干常见光滑类及非标准光滑类在极小极大最优MISE速率中的相变现象，而且可在非参数回归问题之外具有独立的研究价值。