It is a widely observed phenomenon in nonparametric statistics that rate-optimal estimators 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 $\beta$-H\"older smooth regression function f. It is shown that an 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 $\gtrsim \psi_n.$ This proves that any estimator achieving the minimax optimal pointwise estimation rate $\psi_n$ must necessarily balance worst-case bias and worst-case mean absolute deviation. To derive the result, we establish an abstract inequality relating the change of expectation for two probability measures to the mean absolute deviation.

翻译：在非参数统计中，一个广泛观察到的现象是，速率最优的估计器能够平衡偏倚与随机误差。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的速率最优估计器？本文考虑高斯白噪声模型中关于β-赫尔德光滑回归函数f的点估计问题。研究表明，一个具有最坏情形偏倚≲n^{-β/(2β+1)}=: ψ_n的估计器，其最坏情形平均绝对偏差必然有一个下界≳ψ_n。这证明了任何达到极小化最优点估计速率ψ_n的估计器必须平衡最坏情形偏倚与最坏情形平均绝对偏差。为推导该结果，我们建立了一个关于两个概率测度期望变化与平均绝对偏差之间关系的抽象不等式。