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- Title
Data depth for the uniform distribution.
- Authors
Silva, Pedro; Cerdeira, J.; Martins, M.; Monteiro-Henriques, T.
- Abstract
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.
- Subjects
HYPERSPHERICAL method; UNIFORM distribution (Probability theory); MEAN value theorems; RANDOM variables; ECOLOGICAL niche; EUCLIDEAN geometry
- Publication
Environmental & Ecological Statistics, 2014, Vol 21, Issue 1, p27
- ISSN
1352-8505
- Publication type
Article
- DOI
10.1007/s10651-013-0242-7