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.
Highlights
Expectiles were first introduced by Newey and Powell [36] in the context of linear regression as the solution to a minimization problem
Two appendices are placed at the end of the manuscript, Appendix A contains the description of two algorithms, one for the computation of the set of extreme points of the expectile regions of a bivariate dataset, and the other for the computation of the expectile depth, while Appendix B contains the proofs of some mathematical results
In Appendix A.2, we describe an algorithm to compute the bivariate expectile depth that takes advantage of the fact that the angle γ for which u = is the minimizer of expression (17) is π /2 radians distant from the angle that forms the positive horizontal semiaxis and the ray from the origin containing a data point
Summary
Expectiles were first introduced by Newey and Powell [36] in the context of linear regression as the solution to a minimization problem They were so named because they resemble the quantiles of a random variable, but unlike them, they are based on a quadratic loss function, as it is the case of the expectation. Other multivariate generalizations of the expectiles have been considered by Breckling and Chambers [3], who proposed a class of multivariate M-quantiles as the solution to minimization problems similar to those giving rise to quantiles and expectiles Since some of those M-quantiles lie out of the convex hull of the dataset they were built from, in a posterior paper Breckling et al [4] presented an alternative definition of multivariate M-quantiles which still lack to be. Two appendices are placed at the end of the manuscript, Appendix A contains the description of two algorithms, one for the computation of the set of extreme points of the expectile regions of a bivariate dataset, and the other for the computation of the expectile depth, while Appendix B contains the proofs of some mathematical results
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