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, n otions of "more scattered" based on the "halfspace" depth function are show n to generalize versions introduced by P. J. Bickel and E. L. Lehmann (1976 , 1979) in the univariate case and by M. L. Eaten (1982) and H. Oja (1983) in the multivariate cast. Scatter measures are also discussed. with emphasi s on those based on the halfspace depth. Basic desirable properties establi shed 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 multivar iate cases. Further, some properties unique to the depth-based notions are established. (C) 2000 Academic Press.