  In this paper, we propose self-tuned robust estimators for estimating the
mean of heavy-tailed distributions, which refer to distributions with only
finite variances. Our approach introduces a new loss function that considers
both the mean parameter and a robustification parameter. By jointly optimizing
the empirical loss function with respect to both parameters, the
robustification parameter estimator can automatically adapt to the unknown data
variance, and thus the self-tuned mean estimator can achieve optimal
finite-sample performance. Our method outperforms previous approaches in terms
of both computational and asymptotic efficiency. Specifically, it does not
require cross-validation or Lepski's method to tune the robustification
parameter, and the variance of our estimator achieves the Cram\'er-Rao lower
bound. Project source code is available at
\url{https://github.com/statsle/automean}.
