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

Given data in $\mathbb{R}^{p}$, a Tukey $\tau$-trimmed region is the set of all points that have at least Tukey depth $\tau$ in the data. As they are visual, affine equivariant and robust, Tukey trimmed regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and can be interpreted as $p$-variate quantiles, their practical application is impeded by the lack of efficient computational procedures in dimension $p>2$. We construct two algorithms that exploit certain combinatorial properties of the regions. Both calculate the exact region. They run much faster and require substantially less RAM than existing ones.