 Influence function of the sample median

Let $F$ have a unique median $m$ and a continuous probability density function $f$ with $f(m)>0$. The influence function of a quantile at $p=1/2$ givesFor contamination exactly at $m$, the generalized-inverse median remains $m$ and the influence function is zero. Thus its value is determined everywhere by taking $\mathrm{sgn}(0)=0$.

The response is bounded by $1/(2f(m))$, so the gross-error sensitivity is finite. It has nonzero constant tails, giving an infinite rejection point of an influence function, and its jump gives infinite local-shift sensitivity. For a standard normal distribution, the bound is $\sqrt{\pi /2}$. These three different sensitivity measures describe different aspects of the same estimator.