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

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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.

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