The random Tukey depth

The computation of the Tukey depth, also called halfspace depth, is very demanding, even in low dimensional spaces, because it requires that all possible one-dimensional projections be considered. A random depth which approximates the Tukey depth is proposed. It only takes into account a finite number of one-dimensional projections which are chosen at random. Thus, this random depth requires a reasonable computation time even in high dimensional spaces. Moreover, it is easily extended to cover the functional framework. Some simulations indicating how many projections should be considered depending on the kind of problem, sample size and dimension of the sample space among others are presented. It is noteworthy that the random depth, based on a very low number of projections, obtains results very similar to those obtained with the Tukey depth.