Generalization of the simplicial depth: no vanishment outside the convex hull of the distribution support
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The simplicial depth, like other relevant multivariate statistical data depth functions, vanishes right outside the convex hull of the support of the distribution with respect to which the depth is computed. This is problematic when it is required to differentiate among points outside the convex hull of the distribution support, with respect to which the depth is computed, based on their depth values. We provide the first two proposals to overcome this issue as well as several corresponding estimators, which do not vanish right outside the convex hull of the data. The properties of the proposals and of the corresponding estimators are studied, theoretically and by means of Monte Carlo simulations. Furthermore, the proposed methodology can be generally applied and it is here extended to the other most well-know instance of depth: the halfspace, or Tukey, depth, which suffers from the same problematic.
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