  We consider the problem of mean estimation assuming only finite variance. We
study a new class of mean estimators constructed by integrating over random
noise applied to a soft-truncated empirical mean estimator. For appropriate
choices of noise, we show that this can be computed in closed form, and
utilizing relative entropy inequalities, these estimators enjoy deviations with
exponential tails controlled by the second moment of the underlying
distribution. We consider both additive and multiplicative noise, and several
noise distribution families in our analysis. Furthermore, we empirically
investigate the sensitivity to the mean-standard deviation ratio for numerous
concrete manifestations of the estimator class of interest. Our main take-away
is that an inexpensive new estimator can achieve nearly sub-Gaussian
performance for a wide variety of data distributions.
