= 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 $p=1/2$ gives
$$
\operatorname{IF}(x;m,F)=\frac{\operatorname{sgn}(x-m)}{2f(m)}\qquad(x\ne m).
$$
For contamination exactly at $m$, the generalized-inverse <median> remains $m$ and the <influence function> is zero. Thus its value is determined everywhere by taking $\operatorname{sgn}(0)=0$.
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>.
Back to article page
