Robust statistics

Robust statistics studies procedures whose behavior remains controlled under outliers and deviations from an assumed model.

The influence function is the derivative at zero contamination of a statistical functional $T((1-\epsilon )F+\epsilon {\delta }_x)$.

Gross-error sensitivity is the supremum of the absolute influence function over contamination points.

At a distribution with positive density $f$ at its median $m$, the median influence function is $\mathrm{sgn}(x-m)/(2f(m))$ away from $m$.

The finite-sample replacement breakdown point is the largest replacement fraction under which an estimator remains bounded over arbitrary replacement values.

The Huber location estimator solves ${\sum }_i{\psi }_k(x_i-\theta )=0$, where ${\psi }_k$ clips its argument to $[-k,k]$.

The Huber loss is quadratic near zero and linear beyond a fixed cutoff; its derivative is the clipped Huber score.

The median-of-means estimator partitions a sample, computes each group mean, and returns the median of those means.

For an odd sample size $n$ from a distribution with positive continuous density $f$ at its median $m$,