Data depth for the uniform distribution

• DOI: 10.1007/s10651-013-0242-7
• Publication Date: 2013-03-23T07:10:01Z
• Authors: Pedro C. Silva; J. Orestes Cerdeira; M. João Martins; T. Monteiro-Henri...
• Keywords: 0101 mathematics; 01 natural sciences

Given a set $$X$$ of $$k$$ points and a point $$z$$ in the $$n$$ -dimensional euclidean space, the Tukey depth of $$z$$ with respect to $$X$$ , is defined as $$m/k$$ , where $$m$$ is the minimum integer such that $$z$$ is not in the convex hull of some set of $$k-m$$ points of $$X$$ . If $$z$$ belongs to the closed region $$B$$ delimited by an ellipsoid, define the continuous depth of $$z$$ with respect to $$B$$ as the quotient $$V(z)/\text{ Vol }(B)$$ , where $$V(z)$$ is the minimum volume of the intersection of $$B$$ with the halfspaces defined by any hyperplane passing through $$z$$ , and $$\text{ Vol }(B)$$ is the volume of $$B$$ . We consider $$z$$ a random variable and prove that, if $$z$$ is uniformly distributed in $$B$$ , the continuous depth of $$z$$ with respect to $$B$$ has expected value $$1/2^{n+1}$$ . This result implies that if $$z$$ and $$X$$ are uniformly distributed in $$B$$ , the expected value of Tukey depth of $$z$$ with respect to $$X$$ converges to $$1/2^{n+1}$$ as the number of points $$k$$ 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.