In nonparametric statistics, rate-optimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with regression function $f$ in a class of $\beta$-H\"older smooth functions. Let 'worst-case' refer to the supremum over all functions $f$ in the H\"older class. It is shown that any estimator with worst-case bias $\lesssim n^{-\beta/(2\beta+1)}=: \psi_n$ must necessarily also have a worst-case mean absolute deviation that is lower bounded by $\gtrsim \psi_n.$ To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.

翻译：在非参数统计中，速率最优估计量通常需要在偏差与随机误差之间取得平衡。近期关于过参数化的研究提出了一个问题：是否存在不遵循这种权衡关系的速率最优估计量？本文考虑高斯白噪声模型中的逐点估计问题，其中回归函数$f$属于$\beta$-赫尔德光滑函数类。令“最坏情况”表示对赫尔德类中所有函数$f$取上确界。研究表明：任何最坏情况偏差$\lesssim n^{-\beta/(2\beta+1)}=: \psi_n$的估计量，其最坏情况平均绝对偏差必然满足下界$\gtrsim \psi_n$。为推导该结果，我们建立了两个概率分布的期望变化与平均绝对偏差之间关系的抽象不等式。