Fast computation of Tukey trimmed regions and median in dimension $p>2$

Given data in $\mathbb{R}^{p}$, a Tukey $\kappa$-trimmed region, shortly Tukey $\kappa$-region or just Tukey region, is the set of all points that have at least Tukey depth $\kappa$ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical application is impeded by the lack of efficient computational procedures in dimension $p > 2$. We derive a strict bound on the number of facets of a Tukey region and construct a new efficient algorithm to compute the region, which runs much faster than existing ones. The new algorithm is compared with a slower exact algorithm, yielding always the same correct results. Finally, the approach is extended to an algorithm that efficiently calculates the innermost Tukey region and its barycenter, the Tukey median.