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Central regions and dependency


Author Info

  • K. Mosler

    (Universität zu Köln)


The paper introduces an approach to the ordering of dependence which is based on central regions. A d-variate probability distribution is described by a nested family of sets, called central regions. Those regions are affine equivariant, compact and starshaped and concentrate about a properly defined center. They can be seen as level sets of a depth function. Special cases are Mahalanobis, zonoid, and likelihood regions. A d-variate distribution is called more dependent than another one if the volume of each central region is smaller with the first distribution. This dependence order is characterized by an inequality between determinants of certain parameter matrices if either (i) F and G are arbitrary distributions and the central regions are Mahalanobis or (ii) F and G belong to an elliptical family of distributions and the central regions are arbitrary. If the regions are zonoid regions, the dependence order implies the ordering of lift zonoid volumes. Alternatively, the dependence order is applied to the copulae of the given distributions. Generalized correlation indices are proposed which are increasing with the dependence orders.

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Bibliographic Info

Paper provided by EconWPA in its series Econometrics with number 0309004.

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Length: 16 pages
Date of creation: 15 Sep 2003
Date of revision:
Handle: RePEc:wpa:wuwpem:0309004

Note: Type of Document - Post Script; pages: 16 ; figures: included. This is a preprint of an article accepted for publication in "Methology and Computing in Applied Probability"
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Related research

Keywords: Dependence order; generalized correlation; lift zonoid volume; data depth; trimmed regions;

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  1. Dall'Aglio, Marco & Scarsini, Marco, 2001. "When Lorenz met Lyapunov," Statistics & Probability Letters, Elsevier, vol. 54(1), pages 101-105, August.
  2. Koshevoy, G. A. & Mosler, K., 1997. "Multivariate Gini Indices," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 252-276, February.
  3. Hannu Oja, 1999. "Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 26(3), pages 319-343.
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Cited by:
  1. Lange, Tatjana & Mosler, Karl & Mozharovskyi, Pavlo, 2012. "Fast nonparametric classification based on data depth," Discussion Papers in Statistics and Econometrics 1/12, University of Cologne, Department for Economic and Social Statistics.
  2. Ignacio Cascos & Ilya Molchanov, 2006. "Multivariate risks and depth-trimmed regions," Papers math/0606520,, revised Nov 2006.
  3. Anirvan Chakraborty & Probal Chaudhuri, 2014. "On data depth in infinite dimensional spaces," Annals of the Institute of Statistical Mathematics, Springer, vol. 66(2), pages 303-324, April.
  4. Ignacio Cascos & Ilya Molchanov, 2006. "Multivariate Risks And Depth-Trimmed Regions," Statistics and Econometrics Working Papers ws063815, Universidad Carlos III, Departamento de Estadística y Econometría.
  5. Daniek Kosiorowski, 2008. "Scale curve – a robust and nonparametric approach to study a dispersion and interdependence of multivariate distributions," Operations Research and Decisions, Wroclaw University of Technology, Institute of Organization and Management, vol. 4, pages 47-60.
  6. Ignacio Cascos, 2006. "The Expected Convex Hull Trimmed Regions Of A Sample," Statistics and Econometrics Working Papers ws066919, Universidad Carlos III, Departamento de Estadística y Econometría.
  7. Barry C. Arnold, 2005. "Inequality measures for multivariate distributions," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 317-327.
  8. Ilga Molchanov & Michael Schmutz & Kaspar Stucki, 2012. "Invariance properties of random vectors and stochastic processes based on the zonoid concept," Statistics and Econometrics Working Papers ws122014, Universidad Carlos III, Departamento de Estadística y Econometría.
  9. Asis Banerjee, 2014. "A multidimensional Lorenz dominance relation," Social Choice and Welfare, Springer, vol. 42(1), pages 171-191, January.
  10. Beresteanu, Arie & Molinari, Francesca, 2006. "Asymptotic Properties for a Class of Partially Identified Models," Working Papers 06-04, Duke University, Department of Economics.
  11. Nedret Billor & Asheber Abebe & Asuman Turkmen & Sai Nudurupati, 2008. "Classification Based on Depth Transvariations," Journal of Classification, Springer, vol. 25(2), pages 249-260, November.
  12. Karl Mosler, 2004. "Restricted Lorenz dominance of economic inequality in one and many dimensions," Journal of Economic Inequality, Springer, vol. 2(2), pages 89-103, August.
  13. Chiara GIGLIARANO & Karl MOSLER, 2009. "Measuring middle-class decline in one and many attributes," Working Papers 333, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
  14. Marco Dall’Aglio & Marco Scarsini, 2000. "Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex," ICER Working Papers - Applied Mathematics Series 27-2003, ICER - International Centre for Economic Research, revised Jul 2003.
  15. Liesa Denecke & Christine Müller, 2014. "Consistency of the likelihood depth estimator for the correlation coefficient," Statistical Papers, Springer, vol. 55(1), pages 3-13, February.
  16. Gleb A. Koshevoy & Karl Mosler, 2007. "Multivariate Lorenz dominance based on zonoids," AStA Advances in Statistical Analysis, Springer, vol. 91(1), pages 57-76, March.


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