TY - JOUR

T1 - Influence functions for a general class of depth-based generalized quantile functions

AU - Wang, Jin

AU - Serfling, Robert

N1 - Funding Information:
Constructive suggestions by an Associate Editor and a referee are very much appreciated and have greatly improved the paper. Support by National Science Foundation Grant DMS-0103698 is also gratefully acknowledged.

PY - 2006/4

Y1 - 2006/4

N2 - Given a multivariate probability distribution F, a corresponding depth function orders points according to their "centrality" in the distribution F. One useful role of depth functions is to generate two-dimensional curves for convenient and practical description of particular features of a multivariate distribution, such as dispersion and kurtosis. Here the robustness of sample versions of such curves is explored via the influence function approach applied to the relevant functionals, using structural representations of the curves as generalized quantile functions. In particular, for a general class of so-called Type D depth functions including the well-known Tukey or halfspace depth, we obtain influence functions for the depth function itself, the depth distribution function, the depth quantile function, and corresponding depth-based generalized quantile functions. Robustness behavior similar to the usual univariate quantiles is found and quantified: the influence functions are of step function form with finite gross error sensitivity but infinite local shift sensitivity. Applications to a "scale" curve, a Lorenz curve for "tailweight", and a "kurtosis" curve are treated. Graphical illustrations are provided for the influence functions of the scale and kurtosis curves in the case of the bivariate standard normal distribution and the halfspace depth function.

AB - Given a multivariate probability distribution F, a corresponding depth function orders points according to their "centrality" in the distribution F. One useful role of depth functions is to generate two-dimensional curves for convenient and practical description of particular features of a multivariate distribution, such as dispersion and kurtosis. Here the robustness of sample versions of such curves is explored via the influence function approach applied to the relevant functionals, using structural representations of the curves as generalized quantile functions. In particular, for a general class of so-called Type D depth functions including the well-known Tukey or halfspace depth, we obtain influence functions for the depth function itself, the depth distribution function, the depth quantile function, and corresponding depth-based generalized quantile functions. Robustness behavior similar to the usual univariate quantiles is found and quantified: the influence functions are of step function form with finite gross error sensitivity but infinite local shift sensitivity. Applications to a "scale" curve, a Lorenz curve for "tailweight", and a "kurtosis" curve are treated. Graphical illustrations are provided for the influence functions of the scale and kurtosis curves in the case of the bivariate standard normal distribution and the halfspace depth function.

KW - Dispersion

KW - Generalized quantiles

KW - Kurtosis

KW - Multivariate analysis

KW - Nonparametric methods

KW - Robustness

UR - http://www.scopus.com/inward/record.url?scp=33644678116&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644678116&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2005.07.002

DO - 10.1016/j.jmva.2005.07.002

M3 - Article

AN - SCOPUS:33644678116

VL - 97

SP - 810

EP - 826

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 4

ER -