 Tukey median

A Tukey median maximizes halfspace depth. For a probability measure $P$ on ${\mathbb{R}}^d$, the depth of a point $x$ isFor a sample, $P$ is replaced by its empirical distribution, so $P(H)$ is the fraction of observations in the closed half-space $H$. Every direction is considered; choosing a single coordinate direction would give a different procedure.

In one dimension, the depth is the smaller of the masses in the closed rays to the left and right of $x$. For the sample $(0,1,2,3,100)$, point $2$ has depth $3/5$, while $100$ has depth $1/5$. The maximizing point is the sample median. In higher dimensions this gives a geometric extension of a central location that depends on how many observations lie on each side, rather than on their squared distances. Maximizers need not be unique, so a single-valued estimator needs a selection rule.