Trimmed mean

For $0\le \alpha <1/2$, an $\alpha$-trimmed mean sorts $n$ observations, removes the smallest and largest $k=\lfloor \alpha n\rfloor$, and averages the remaining $n-2k$ values. It limits the effect of a small number of extreme observations while still averaging a central portion of the data.

For $(1,2,3,4,100)$ and $\alpha =1/5$, removing $1$ and $100$ gives trimmed mean $(2+3+4)/3=3$, compared with sample mean $22$. Replacing $100$ by a larger value leaves this trimmed mean unchanged. Replacing enough observations can still move the retained central values arbitrarily far.

For a probability distribution with quantile function $F^{-1}$, the population statistical functional isThe normalizing denominator is the retained fraction of probability. At a regular symmetric reference probability distribution, the influence function of a trimmed mean is a clipped version of the observation, scaled by this fraction. Centering and symmetry assumptions matter when using that simpler expression.

At a differentiable distribution symmetric about zero, put $k=-F^{-1}(\alpha )$. The influence function of the $\alpha$-trimmed mean is