Multivariate analysis by data depth: descriptive statistics, graphics and inference, (with discussion and a rejoinder by Liu and Singh)

TLDR

Data depth quantifies how far a multivariate observation lies from the center of its distribution, yielding a natural center‑outward ordering of sample points. The authors introduce DD‑plots and new diagnostic tools to assess multivariate normality and serve as graphical inference aids. Using the depth ordering, they develop quantitative and graphical methods—including sunburst plots, DD‑plots, and dispersion‑ratio diagnostics—to analyze location, scale, bias, skewness, kurtosis, and to compare inference procedures. These methods produce one‑dimensional curves that are easy to visualize, and the affine invariance of data depth ensures that the resulting statistics and plots inherit appropriate invariance properties.

Abstract

A data depth can be used to measure the “depth” or “outlyingness” of a given multivariate sample with respect to its underlying distribution. This leads to a natural center-outward ordering of the sample points. Based on this ordering, quantitative and graphical methods are introduced for analyzing multivariate distributional characteristics such as location, scale, bias, skewness and kurtosis, as well as for comparing inference methods. All graphs are one-dimensional curves in the plane and can be easily visualized and interpreted. A “sunburst plot” is presented as a bivariate generalization of the box-plot. DD-(depth versus depth) plots are proposed and examined as graphical inference tools. Some new diagnostic tools for checking multivariate normality are introduced. One of them monitors the exact rate of growth of the maximum deviation from the mean, while the others examine the ratio of the overall dispersion to the dispersion of a certain central region. The affine invariance property of a data depth also leads to appropriate invariance properties for the proposed statistics and methods.