Computational Statistics

, Volume 28, Issue 4, pp 1853–1880 | Cite as

Inference for vast dimensional elliptical distributions

Original Paper

Abstract

We propose a quantile–based method to estimate the parameters of an elliptical distribution, and a battery of tests for model adequacy. The method is suitable for vast dimensions as the estimators for location and dispersion have closed–form expressions, while estimation of the tail index boils down to univariate optimizations. The tests for model adequacy are for the null hypothesis of correct specification of one or several level contours. A Monte Carlo study to three distributions (Gaussian, Student–t and elliptical stable) for dimensions 20, 200 and 2000 reveals the goodness of the method, both in terms of computational time and finite samples. An empirical application to financial data illustrates the method.

Keywords

Quantiles Elliptical family Simulations Heavy tails 

References

  1. Babu GJ, Rao CR (1988) Joint asymptotic distribution of marginal quantiles and quantile functions in sample from a multivariate population. J Multivar Anal 27:15–23MathSciNetMATHCrossRefGoogle Scholar
  2. Barigozzi M, Halbleib R, Veredas D (2012) Which model to match? ECARES WP 2012/04Google Scholar
  3. Bingham NH, Kiesel R (2003) Semi-parametric modelling in finance: theoretical foundations. Quant Financ 2:241–250MathSciNetCrossRefGoogle Scholar
  4. Bingham NH, Kiesel R, Schmidt R (2003) A semi-parametric approach to risk management. Quant Financ 3:426–441MathSciNetCrossRefGoogle Scholar
  5. Cambais S, Huang S, Simons G (1981) On the theory of elliptically contoured distributions. J Multivar Anal 11:368–385CrossRefGoogle Scholar
  6. Chambers JM, Mallows CL, Stuck BW (1976) A method for simulating stable random variables. J Am Stat Assoc 71, 340–344. Corrections 82 (1987):704, 83 (1988):581Google Scholar
  7. Chen XC, Jacho-Chavez DT, Linton O (2009) An alternative way of computing efficient instrumental variable estimators. LSE STICERD Research Paper EM/2009/536Google Scholar
  8. Cramér H (1946) Mathematical methods of statistics. Princeton University Press, PrincetonMATHGoogle Scholar
  9. de Vries CG (1991) On the relation between GARCH and stable processes. J Econom 48:313–324MATHCrossRefGoogle Scholar
  10. Dominicy Y, Veredas D (2012) The method of simulated quantiles. J Econ, forthcomingGoogle Scholar
  11. Fama E, Roll R (1971) Parameter estimates for symmetric stable distributions. J Am Stat Assoc 66:331–338MATHCrossRefGoogle Scholar
  12. Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions. Chapman and Hall, New YorkMATHGoogle Scholar
  13. Frahm G (2004) Generalized elliptical distributions. PhD thesis, University of CologneGoogle Scholar
  14. Ghose D, Kroner KF (1995) The relationship between GARCH and symmetric stable processes: finding the source of fat tails in financial data. J Empir Financ 2:225–251CrossRefGoogle Scholar
  15. Gonzalez-Rivera G, Senyuz Z, Yoldas E (2011) Autocontours: dynamic specification testing. J Bus Econ Stat 29:186–200MathSciNetMATHCrossRefGoogle Scholar
  16. Gonzalez-Rivera G, Yoldas E (2011) Autocontour-based evaluation of multivariate predictive densities, Int J Forecast, forthcomingGoogle Scholar
  17. Gouriéroux C, Monfort A, Renault E (1993) Indirect inference. J Appl Econom 8:85–118CrossRefGoogle Scholar
  18. Hallin M, Paindaveine D, Oja H (2006) Semiparametrically efficient rank-based inference for shape. II. Optimal R-estimation of shape. Ann Stat 34:2757–2789MathSciNetMATHCrossRefGoogle Scholar
  19. Hautsch N, Kyj LM, Oomen RC (2011) A blocking and regularization approach to high dimensional realized covariance estimation, forthcoming in the J Appl EconGoogle Scholar
  20. Hill JB, Renault E (2012) Generalized method of moments with tail trimming. University of North Carolina at Chapel Hill, MimeoGoogle Scholar
  21. Kelker D (1970) Distribution theory of spherical distributions and a location-scale parameter generalization. Sankhya A32:419–430MathSciNetGoogle Scholar
  22. Laloux L, Cizeau P, Bouchaud J-P, Potters M (1999) Noise dressing of financial correlation matrices. Phys Rev Lett 83:1467–1470CrossRefGoogle Scholar
  23. Laurent S, Veredas D (2011) Testing conditional asymmetry. A residual-based approach, J Econ Dyn Control, forthcomingGoogle Scholar
  24. Lombardi MJ, Veredas D (2009) Indirect inference of elliptical fat tailed distributions. Comput Stat Data Anal 53:2309–2324MathSciNetMATHCrossRefGoogle Scholar
  25. McCulloch JH (1986) Simple consistent estimators of stable distribution parameters. Commun Stat Simul Comput 15(4):1109–1136MathSciNetMATHCrossRefGoogle Scholar
  26. Nolan JP (2010) Multivariate elliptically contoured stable distributions: theory and estimation. MimeoGoogle Scholar
  27. Rosadi D, Deistler M (2011) Estimating the codifference function of linear time series models with infinite variance. Metrika 73:395–429MathSciNetMATHCrossRefGoogle Scholar
  28. Tola V, Lillo F, Gallegati M, Mantegna R (2008) Cluster analysis for portfolio optimization. J Econ Dyn Control 32:235–258MathSciNetMATHCrossRefGoogle Scholar
  29. Tyler DE (1987) A distribution-free M-estimator of multivariate scatter. Ann Stat 15:234–251MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.ECARES, Solvay Brussels School of Economics and ManagementUniversité libre de Bruxelles BrusselsBelgium
  2. 2.School of International Liberal StudiesWaseda UniversityTokyoJapan

Personalised recommendations