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Data depth for the uniform distribution
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DOI：10.1007/s10651-013-0242-7.png)
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Given a set of points and a point in the -dimensional euclidean space, the Tukey depth of with respect to , is defined as , where is the minimum integer such that is not in the convex hull of some set of points of . If belongs to the closed region delimited by an ellipsoid, define the continuous depth of with respect to as the quotient , where is the minimum volume of the intersection of with the halfspaces defined by any hyperplane passing through , and is the volume of . We consider a random variable and prove that, if is uniformly distributed in , the continuous depth of with respect to has expected value . This result implies that if and are uniformly distributed in , the expected value of Tukey depth of with respect to converges to as the number of points goes to infinity. These findings have applications in ecology, namely within the niche theory, where it is useful to explore and characterize the distribution of points inside species niche.
Keyword:
Hyperspherical cap
Mean value
Species niche
Tukey depth
Uniform distribution
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