Influence function

Summary

The influence function of a statistical functional measures the effect on the functional of adding an infinitesimal amount of contamination at a point $x$. It is a directional derivative of the functional in the direction of a point mass, a Gateaux derivative. A bounded influence function means that no single observation can have a large effect, and the supremum of its absolute value is the gross-error sensitivity. Under regularity conditions it also gives the asymptotic variance of the estimator, so it links robustness and efficiency.