Expectile depth: Theory and computation for bivariate datasets Articles uri icon

publication date

  • July 2021

start page

  • 1

end page

  • 17

volume

  • 184

International Standard Serial Number (ISSN)

  • 0047-259X

Electronic International Standard Serial Number (EISSN)

  • 1095-7243

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.

keywords

  • algorithm; bagplot; data depth; depth region; expectile