Publication: Expectile depth: Theory and computation for bivariate datasets
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2021-07-01
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Elsevier
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Abstract
Expectiles are the solution to an asymmetric least squares minimization problem for
univariate data. They resemble the quantiles, and just like them, expectiles are indexed
by a level α in the unit interval. In the present paper, we introduce and discuss the main
properties of the (multivariate) expectile regions, a nested family of sets, whose instance
with level 0 < α ≤ 1/2 is built up by all points whose univariate projections lie between
the expectiles of levels α and 1 − α of the projected dataset. Such level is interpreted
as the degree of centrality of a point with respect to a multivariate distribution and
therefore serves as a depth function. We propose here algorithms for determining all
the extreme points of the bivariate expectile regions as well as for computing the depth
of a point in the plane. We also study the convergence of the sample expectile regions to
the population ones and the uniform consistency of the sample expectile depth. Finally,
we present some real data examples for which the Bivariate Expectile Plot (BExPlot) is
introduced.
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Keywords
Algorithm, Bagplot, Data depth, Depth Region, Expectile
Bibliographic citation
Cascos, I., & Ochoa, M. (2021). Expectile depth: Theory and computation for bivariate datasets. In Journal of Multivariate Analysis, 184, p. 104757.