Improved Concentration for Mean Estimators via Shrinkage

We study a class of robust mean estimators $\widehat{\mu }$ obtained by adaptively shrinking the weights of sample points far from a base estimator $\widehat{\kappa }$. Given a data-dependent scaling factor $\widehat{\alpha }$ and a weighting function $w:[0,\mathrm{\infty })\to [0,1]$, we let $\widehat{\mu }=\widehat{\kappa }+\frac{1}{n}{\sum }_{i=1}^n(X_i-\widehat{\kappa })w(\widehat{\alpha }\mathrm{\mid }{X}_i-\widehat{\kappa }\mathrm{\mid })$. We prove that, under mild assumptions over $w$, these estimators achieve stronger concentration bounds than the base estimate $\widehat{\kappa }$, including sub-Gaussian guarantees. This framework unifies and extends several existing approaches to robust mean estimation in $\mathbb{R}$. Through numerical experiments, we show that our shrinking approach translates to faster concentration, even for small sample sizes.