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UCU T, where p is the generalized variance, the orthogonal matrix U contains the eigenvectors and C is the diagonal matrix of standardized eigenvalues Z Z.. Z. det C 1. As in Bensmail and Celeux 1996, we use the terms scale, shape and orientation for items, C and U. If z comes from a spherical distribution with the location vector 0 and covariance matrix I, then y UC1 2 1 2z is elliptically symmetric with the location vector, scale, shape C and orientation U. Our plan is to first define a multivariate centered rank vector. This vector, in many ways, represents an extension of the idea of a univariate rank. In addition, it has certain nice affine equivariance properties. We only provide a Z. Z. sketch here; see Hettmansperger, Mottonen and Oja 1998 or Oja 1999 for ¨ ¨ details. We then consider the rank covariance matrix, RCM. Visuri, Koivunen Z. and Oja 1999 show that if the standardized eigenvalues and the eigenvectors of the covariance matrix are c c and u,..., u, respectively, 1 p 1 p then c 1 c 1 and u,..., u are the standardized eigenvalues and 1 p 1 p the eigenvectors for the theoretical RCM. The sample RCM is more robust than the sample covariance matrix and, hence, provides a robust estimate of the underlying shape and orientation for the elliptical distribution. This, along with a robust estimate of Wilk's generalized variance, can be used to robustly estimate. However, here we use only the standardized eigenvalues and the eigenvectors to define a robust version of depth. We next sketch the construction of the rank vector and corresponding sample RCM. We begin with p-dimensional data x,..., x. The volume of 1 n the p-variate simplex determined by x and p observation vectors with indices i i is 1 p
, shape C or orientation U. The log scale facilitates comparison of scale near the centers. Compare Z. these plots to Figure 7 a, b in the paper. The other nice application discussed by the authors is for the comparison of scatter of the multivariate estimates Z. of location; see Figure 8 a, b, c in the paper. The comparison based on ellipses would be quite natural here since, typically, the estimators will have multivariate normal limiting distributions. Another way to compare scales for two distributions is to look at a PP-plot of the elliptical areas for the two samples. Essentially, it is a plot of the empirical cdf's of the elliptical areas determined by the data in each sample. Z. Z. Figure 3 shows a PP-scale plot of A versus D. Z. Note that beyond 0.5 the empirical cdf's of the elliptical areas, F u A Z. Z. Z. F u, indicating that D has more scatter or larger scale than A. The area D under the curve could provide a measure and, hence, in the elliptical case, an asymptotically distribution-free test for scale differences. The test statistic then is the Mann Whitney Wilcoxon U-statistic calculated from the depths. In the univariate case, this corresponds to a rank test based on magnitudes of the centered observations. In the comparison in Figure 4, the observed Z. p-value one-sided test is 0.22.
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