The random Tukey depth
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The computation of the Tukey depth, also called halfspace depth, is very demanding, even in low dimensional spaces, because it requires the consideration of all possible one-dimensional projections. In this paper we propose a random depth which approximates the Tukey depth. It only takes into account a finite number of one-dimensional projections which are chosen at random. Thus, this random depth requires a very small computation time even in high dimensional spaces. Moreover, it is easily extended to cover the functional framework.
We present some simulations indicating how many projections should be considered depending on the sample size and on the dimension of the sample space. We also compare this depth with some others proposed in the literature. It is noteworthy that the random depth, based on a very low number of projections, obtains results very similar to those obtained with other depths.