Last week, we've introduced the concept of exchangeable variables, i.e. satisfying http://freakonometrics.blog.free.fr/public/perso5/ellex01.gif for any matrix  http://freakonometrics.blog.free.fr/public/perso5/ellex02.gif , i.e. http://freakonometrics.blog.free.fr/public/perso5/ellexch.gif is a permutation matrix: http://freakonometrics.blog.free.fr/public/perso5/ellexch.gif belongs to the orthogonal group, http://freakonometrics.blog.free.fr/public/perso5/ellex04.gif, and with elements in http://freakonometrics.blog.free.fr/public/perso5/ellex07.gif. It is possible to extend that family, considering all matrices in the orthogonal group, i.e. http://freakonometrics.blog.free.fr/public/perso5/ellex01.gif for all http://freakonometrics.blog.free.fr/public/perso5/ellex03.gif. Since orthogonal matrices can be seen as rotation matrices,

http://freakonometrics.blog.free.fr/public/perso5/ellex06.gif

it will mean, e.g. that density is invariant by rotations,

So level curves will be circles (in dimension 2), or more generally spheres. This will yield the concept of spherical distribution (or spherically contoured distributions), that will be extended to elliptical distributions (see e.g. Hartman & Wintner (1940), Kelker (1970) or Cambanis, Huang & Simons (1979))

  • spherically contoured distributions

http://freakonometrics.blog.free.fr/public/perso5/ellex20.gif has a spherical distribution if its characteristic function can be written

http://freakonometrics.blog.free.fr/public/perso5/ellex21.gif

for some function http://freakonometrics.blog.free.fr/public/perso5/ellex22.gif. Such a function will be called generator, and we say that http://freakonometrics.blog.free.fr/public/perso5/ellex28.gif. Equivalently, http://freakonometrics.blog.free.fr/public/perso5/ellex20.gif has a spherical distribution if http://freakonometrics.blog.free.fr/public/perso5/ellex24.gif. A popular example is the Gaussian distribution (centered, with independent margins)

http://freakonometrics.free.fr/spherical-normal.gif

Note that there exist a nice stochastic representation of spherically contoured distribution, http://freakonometrics.blog.free.fr/public/perso5/ellex10.gif where http://freakonometrics.blog.free.fr/public/perso5/ellex12.gif is a positive random variable, independent of http://freakonometrics.blog.free.fr/public/perso5/ellex11.gif, uniformly distributed over the unit sphere of http://freakonometrics.blog.free.fr/public/perso5/ellex26.gif, i.e.

http://freakonometrics.blog.free.fr/public/perso5/ellex27.gif

This construction can be related to the following decomposition

http://freakonometrics.blog.free.fr/public/perso5/ellex13.gif

  • from circles to ellipses

From that geometric interpretation, and from the connexion with the Gaussian distribution, it is possible to move circles (and spheres) to ellipses. If http://freakonometrics.blog.free.fr/public/perso5/ellex30.gif and http://freakonometrics.blog.free.fr/public/perso5/ellex31.gif for some variance-covariance matrix http://freakonometrics.blog.free.fr/public/perso5/ellex32.gif, then

http://freakonometrics.blog.free.fr/public/perso5/ellex36.gif

where http://freakonometrics.blog.free.fr/public/perso5/ellex35.gif is some square-root of http://freakonometrics.blog.free.fr/public/perso5/ellex32.gif, i.e. http://freakonometrics.blog.free.fr/public/perso5/ellex34.gif. Based on that transformation (through that covariance matrix), level curves of the density are non longer circles, be ellipses,

http://freakonometrics.blog.free.fr/public/perso5/Ell-X.gif

if we change only the variance of the first component (above), while if we change the variance of the second one (below)

http://freakonometrics.blog.free.fr/public/perso5/Ell-Y.gif

If we change only the correlation, the axis of the ellipse are still the first and the second diagonal

http://freakonometrics.blog.free.fr/public/perso5/Ell-R.gif

while the impact of correlation when X and Y do not have the same variance gives us the following transformations,

http://freakonometrics.blog.free.fr/public/perso5/Ell-XR.gif

  • elliptically contoured distributions

A random vector is said to be elliptical, http://freakonometrics.blog.free.fr/public/perso5/ellex37.gif if

http://freakonometrics.blog.free.fr/public/perso5/ellex36.gif

where http://freakonometrics.blog.free.fr/public/perso5/ellex35.gif is some square-root of http://freakonometrics.blog.free.fr/public/perso5/ellex32.gif, i.e. http://freakonometrics.blog.free.fr/public/perso5/ellex34.gif, and http://freakonometrics.blog.free.fr/public/perso5/ellex28.gif. It characteristic function is then

http://freakonometrics.blog.free.fr/public/perso5/ellex38.gif

Further, in that case, http://freakonometrics.blog.free.fr/public/perso5/ellex39.gif while http://freakonometrics.blog.free.fr/public/perso5/ellex40.gif.

http://freakonometrics.free.fr/elliptical-normal.gif

Note that the two most popular elliptical distributions (the Gaussian and Student's t) can be obtained in R as follows,

> library(mnormt)
> x <- seq(-2,4,length=21)
> mu <- c(1,3,2)
> Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
> df <- 4
> x=c(0,0);y=c(0,1); z=c(0,2)
> dmt(cbind(x,y,z), mu, Sigma,df)
[1] 0.006957689 0.020602030
> rmt(n=5, mu, Sigma, df)
[,1]       [,2]      [,3]
[1,]  0.42210352  2.7539135  1.659392
[2,]  1.07968146 -0.1364883  4.851956
[3,] -0.04107115  1.6163407  4.123731
[4,]  0.19784451  2.9329165  1.013374
[5,]  1.13456027  0.4737548 -2.054909