Nonparametric Notions of Multivariate "Scatter Measure" and "More Scattered" Based on Statistical Depth Functions

Nonparametric notions of multivariate "scatter measure" and "more scattered," based on statistical depth functions, are investigated. In particular, notions of "more scattered" based on the "halfspace" depth function are shown to generalize versions introduced by P. J. Bickel and E. L. Lehmann (1976, 1979) in the univariate case and by M. L. Eaton (1982) and H. Oja (1983) in the multivariate case. Scatter measures are also discussed, with emphasis on those based on the halfspace depth. Basic desirable properties established for the previous versions of "more scattered" are shown to carry over to the depth-based notions as well, in both the univariate and the multivariate cases. Further, some properties unique to the depth-based notions are established.