1,720,987 research outputs found
On a multivariate generalization of Gini's mean difference
No full textSerie Redazioni Provvisorie, Dip. di Statistica, Università di Venezia, N. 1 200
Discriminant analysis of von Mises-Fisher distributions, Proceedings of 47th Scientific Meeting of the Italian Statistical Society, S. Cabras, T. Di Battista and W. Racugno,editors. Cagliari, June 10-14, 2014. CUEC Editrice. ISBN 978-88-8467-874-4.
No full textThe influence function of simplicial depth
No full textSerie Redazioni Provvisorie, Dip. Statistica, Università di Venezia, N. 2 200
Data depth and correlation
No full textWe describe depth–based graphical displays that show the interdependence of multivariate distributions. The plots involve one–dimensional curves or bivariate scatterplots, so they are easier to interpret than correlation matrices. The correlation curve, modelled on the scale curve of Liu et al. (1999), compares the volume of the observed central regions with the volume under independence. The correlation DD–plot is the scatterplot of depth values under a reference distribution against depth values under independence. The area of the plot gives a measure of distance from independence. Correlation curve and DD-plot require an lsquoindependencersquo model as a baseline: Besides classical parametric specifications, a nonparametric estimator, derived from the randomization principle, is used. Combining data depth and the notion of quadrant dependence, quadrant correlation trajectories are obtained which allow simultaneous representation of subsets of variables. The properties of the plots for the multivariate normal distribution are investigated. Some real data examples are illustrated
Data depth, random simplices and multivariate dispersion
No full textDepth functions give information not only on the location but also on the dispersion of probability distributions.The Lebesgue integral o fLiu's simplicial depth function is equal
to the expected volume of the random simplex whose vertices are p + 1 independent observations from there levant distribution. Oja'svolume depthisthe Lebesgueintegral
of a linear transformation of the influence function of simplicial depth. The relation of these results with dispersive orderings of distributions is discussed. Some properties of
Mahalanobis' and halfspace depth are illustrated
Nonparametric Multivariate Analysis by Data Depth
No full textSerie Rapporti di Ricerca, Dip. di Statistica, Università di Venezia, N. 1 200
Influence function of halfspace depth
No full textAbstractThe sensitivity of halfspace depth values and contours to perturbations of the underlying distribution is investigated. The influence function of the halfspace depth of any point x∈Rp is bounded and discontinuous; it is constant and positive when the perturbing observation z is placed in any optimal halfspace and it is constant and negative when z is placed in any non-optimal halfspace. When the optimal halfspace is unique a von Mises expansion allows an easy derivation of the asymptotic distribution of the sample halfspace depth. In the sampling case, in general, addition of a single observation outside the convex hull of the sample alters all the depth regions but only the outer region can be arbitrarily expanded. To obtain the same effect on the inner regions the size of the perturbation is required to be not less than the depth orders. Numerical illustrations of the results are given
Discriminant analysis with high dimensional von Mises-Fisher distributions
No full textThis paper extends previous work in discriminant analysis with vMF distributions (e. g., Morris & Laycock, Biometrika, 1974) to general dimension, allowing computation of misclassification probabilities and ROC curves. The key result is the probability distribution of the cosine transformation of a vMF distribution, that is, the random variable Ua = aTX , where X = (X1, ..., Xp)T is a random direction of Sp with vMF distribution and a = (a1, ..., ap)T is a fixed non - random direction of Sp . This transformation is of general importance in multivariate analysis, in particular it underlies discriminant analysis both in the two - group and in the multiple group problem. It allows also to check the surmise that two - group maximum likelihood discriminant rule is equivalent to Fisher's linear discriminant function
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