  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.
