Elliptical distribution

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In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid, respectively, in iso-density plots.

Definition[edit]

Elliptical distributions can be defined using characteristic functions. A multivariate distribution is said to be elliptical if its characteristic function is of the form[1]

for a specified vector , positive-definite matrix , and characteristic function . The function is known as the characteristic generator of the elliptical distribution.[2]

Elliptical distributions can also be defined in terms of their density functions. When they exist, the density functions f have the structure:

where is the scale factor, is an -dimensional random vector with median vector (which is also the mean vector if the latter exists), is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.[3]

Properties[edit]

In the 2-dimensional case, if the density exists, each iso-density locus (the set of x1,x2 pairs all giving a particular value of ) is an ellipse or a union of ellipses (hence the name elliptical distribution). More generally, for arbitrary n, the iso-density loci are unions of ellipsoids. All these ellipsoids or ellipses have the common center μ and are scaled copies (homothets) of each other.

The multivariate normal distribution is the special case in which . While the multivariate normal is unbounded (each element of can take on arbitrarily large positive or negative values with non-zero probability, because for all non-negative ), in general elliptical distributions can be bounded or unbounded—such a distribution is bounded if for all greater than some value.

Note that there exist elliptical distributions that have infinite mean and variance, such as the multivariate Student's t-distribution or the multivariate Cauchy distribution .[4]

Because the variable x enters the density function quadratically, all elliptical distributions are symmetric about

Applications[edit]

Elliptical distributions are used in statistics and in economics.

In mathematical economics, elliptical distributions have been used to describe portfolios in mathematical finance.[5]

Statistics[edit]

In statistics, the multivariate normal distribution was used in classical multivariate analysis for the development of statistical procedures, for example, for estimation and hypothesis-testing. Many classical methods, which are optimal for the normal distribution, continue to have optimal or desirable properties for suitable elliptical distributions, for example such distributions with finite variances.[6][7][8]

Classical methods can be extended to multivariate models in which the mean of the elliptical distribution is a bilinear function (growth curve model). The analysis of such models uses multilinear algebra and the differential calculus of matrices (matrix calculus).[8][9][10][11]

References[edit]

  1. ^ Stamatis Cambanis; Steel Huang; Gordon Simons (1981). "On the Theory of Elliptically Contoured Distributions". Journal of Multivariate Analysis. 11: 368–385. doi:10.1016/0047-259x(81)90082-8. 
  2. ^ Härdle and Simar (2012), p. 178.
  3. ^ Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.
  4. ^ Z. Landsman, E. Valdez, Tail conditional expectations for elliptical distribution North Am. Actuarial J., 7 (4) (2003), pp. 55–71
  5. ^ (Chamberlain 1983; Owen and Rabinovitch 1983)
  6. ^ Anderson (2004)
  7. ^ Arnold (1981)
  8. ^ a b Fang & Zhang (1990)
  9. ^ Pan & Fang (2007)
  10. ^ Kollo & von Rosen (2005)
  11. ^ Magnus & Neudecker (1999)
Bibliography
  • Anderson, T. W. (2004). An introduction to multivariate statistical analysis (3rd ed.). New York: John Wiley and Sons. ISBN 9789812530967. 
  • Arnold, Steven F. (1981). The theory of linear models and multivariate analysis. Wiley. 
  • Chamberlain, G. (1983). "A characterization of the distributions that imply mean-variance utility functions", Journal of Economic Theory 29, 185–201. doi:10.1016/0022-0531(83)90129-1
  • Fang, Kai-Tai; Zhang, Yao-Ting (1990). Generalized multivariate analysis. Science Press (Beijing) and Springer-Verlag (Berlin). ISBN 3540176519. 9783540176510. 
  • Fang, Kai-Tai; Kotz, Samuel; Ng, Kai-Wang (1990). Symmetric multivariate and related distributions. London: Chapman & Hall. 
  • Härdle, Wolfgang Karl; Simar, Lèopold (2012). Applied multivariate statistical analysis (3rd ed.). Springer. 
  • Kollo, Tõnu; von Rosen, Dietrich (2005). Advanced multivariate statistics with matrices. Dordrecht: Springer. ISBN 978-1-4020-3418-3. 
  • Landsman, Zinoviy M.; Valdez, Emiliano A. (2003) Tail Conditional Expectations for Elliptical Distributions (with discussion), The North American Actuarial Journal, 7, 55–123.
  • McNeil, Alexander; Frey, Rüdiger; Embrechts, Paul (2005). Quantitative Risk Management. Princeton University Press. ISBN 0-691-12255-5. 
  • Magnus, Jan R.; Neudecker, Heinz (1999). Matrix differential calculus with applications in statistics and econometrics (Revised ed.). New York: John Wiley & Sons. ISBN 9780471986331. 
  • Owen, J., and Rabinovitch, R. (1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice", Journal of Finance 38, 745–752. JSTOR 2328079
  • Pan, Jianxin; Fang, Kaitai (2007). Growth curve models and statistical diagnostics. Beijing: Science Press. ISBN 9780387950532.