The multinormal distribution belongs to the large family of elliptical distributions
which has recently gained a lot of attention in financial mathematics. Elliptical distributions
are often used, particularly in risk management.
5.4 Spherical and Elliptical Distributions
is said to have a spherical
if its characteristic function
for some scalar function
which is then called
the characteristic generator of the spherical distribution
We will write
This is only one of several possible ways to define spherical distributions.
We can see spherical distributions as an extension of the standard multinormal
The random radius is related to the generator by a relation described
in Fang et al. (1990, p.29).
The moments of
, provided that they exist, can be expressed
in terms of one-dimensional integrals (Fang et al.; 1990).
A spherically distributed random vector does not, in general, necessarily possess a density.
However, if it does, the marginal densities of dimension smaller than are continuous and the marginal
densities of dimension smaller than are differentiable (except possibly at the origin in
Univariate marginal densities for greater than are nondecreasing on and
nonincreasing on .
is said to have an elliptical
has the same distribution as
matrix such that
. We shall write
The elliptical distribution can be seen as an extension of
The multivariate t-distribution.
be independent. The random vector
has a multivariate
degrees of freedom.
-distribution belongs to the family of p-dimensioned spherical distributions.
The multinormal distribution.
shows a density surface of the multivariate normal distribution:
Notice that the density is constant on ellipses. This is the reason for calling
this family of distributions ``elliptical''.