Fast and exact implementation of 3-dimensional Tukey depth regions

Tukey depth regions are important notions in nonparametric multivariate data analysis. A $\tau$-th Tukey depth region ${\mathcal{D}}_{\tau }$ is the set of all points that have at least depth $\tau$. While the Tukey depth regions are easily defined and interpreted as $p$-variate quantiles, their practical applications is impeded by the lack of efficient computational procedures in dimensions with $p>2$. Feasible algorithms are available, but practically very slow. In this paper we present a new exact algorithm for 3-dimensional data. An efficient implementation is also provided. Data examples indicate that the proposed algorithm runs much faster than the existing ones.