The depth function of a population distribution View Full Text


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Article Info

DATE

1999-10

AUTHORS

Peter J. Rousseeuw, Ida Ruts

ABSTRACT

Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a finite data set. Here we focus on the depth function of an arbitrary probability distribution on ℝp, and even of a non-probability measure. The halfspace depth of any point θ in ℝp is the smallest measure of a closed halfspace that contains θ. We review the properties of halfspace depth, enriched with some new results. For various measures, uniform as well as non-uniform, we derive an expression for the depth function. We also compute the Tukey median, which is the θ in which the depth function attains its maximal value. Various interesting phenomena occur. For the uniform distribution on a triangle, a square or any regular polygon, the depth function has ridges that correspond to an 'inversion' of depth contours. And for a product of Cauchy distributions, the depth contours are squares. We also consider an application of the depth function to voting theory. More... »

PAGES

213-244

Journal

TITLE

Metrika

ISSUE

3

VOLUME

49

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/pl00020903

DOI

http://dx.doi.org/10.1007/pl00020903

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1106249797


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