5.4 Spherical and Elliptical Distributions
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.
DEFINITION 5.1
A
![$(p \times 1)$](mvahtmlimg480.gif)
random vector
![$Y$](mvahtmlimg237.gif)
is said to have a spherical
distribution
![$S_p(\phi)$](mvahtmlimg1782.gif)
if its characteristic function
![$\psi_Y(t)$](mvahtmlimg1783.gif)
satisfies:
![$\psi_Y(t) = \phi(t^\top t)$](mvahtmlimg1784.gif)
for some scalar function
![$\phi(.)$](mvahtmlimg1785.gif)
which is then called
the characteristic generator of the spherical distribution
![$S_p(\phi)$](mvahtmlimg1782.gif)
.
We will write
![$Y \sim S_p(\phi)$](mvahtmlimg1786.gif)
.
This is only one of several possible ways to define spherical distributions.
We can see spherical distributions as an extension of the standard multinormal
distribution
.
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
both cases).
Univariate marginal densities for
greater than
are nondecreasing on
and
nonincreasing on
.
DEFINITION 5.2
A
![$(p \times 1)$](mvahtmlimg480.gif)
random vector
![$X$](mvahtmlimg31.gif)
is said to have an elliptical
distribution
with parameters
![$\mu(p\times 1)$](mvahtmlimg1800.gif)
and
![$\Sigma(p \times p)$](mvahtmlimg1801.gif)
if
![$X$](mvahtmlimg31.gif)
has the same distribution as
![$\mu + \data{A}^\top Y$](mvahtmlimg1802.gif)
,
where
![$Y \sim S_k(\phi)$](mvahtmlimg1803.gif)
and
![$\data{A}$](mvahtmlimg319.gif)
is a
![$(k \times p)$](mvahtmlimg1804.gif)
matrix such that
![$\data{A}^\top \data{A}=\Sigma$](mvahtmlimg1805.gif)
with
![$\mathop{\rm {rank}}(\Sigma)=k$](mvahtmlimg1806.gif)
. We shall write
![$X \sim EC_p(\mu,\Sigma,\phi)$](mvahtmlimg1807.gif)
.
REMARK 5.1
The elliptical distribution can be seen as an extension of
![$N_p(\mu,\Sigma)$](mvahtmlimg1282.gif)
.
EXAMPLE 5.6
The multivariate t-distribution.
Let
![$Z \sim N_p(0,\data{I}_{p})$](mvahtmlimg1808.gif)
and
![$s \sim \chi^2_m$](mvahtmlimg1809.gif)
be independent. The random vector
![\begin{equation*}
Y=\sqrt{m} \ \frac{Z}{s}
\end{equation*}](mvahtmlimg1810.gif)
has a multivariate
![$t$](mvahtmlimg672.gif)
-distribution with
![$m$](mvahtmlimg1811.gif)
degrees of freedom.
Moreover the
![$t$](mvahtmlimg672.gif)
-distribution belongs to the family of p-dimensioned spherical distributions.
EXAMPLE 5.7
The multinormal distribution.
Let
![$ X \sim N_p (\mu, \Sigma) $](mvahtmlimg1377.gif)
. Then
![$X \sim EC_p(\mu,\Sigma,\phi)$](mvahtmlimg1807.gif)
and
![$\phi(u)=\exp{(-u/2)}$](mvahtmlimg1812.gif)
.
Figure
4.3 shows a density surface of the multivariate normal distribution:
![$f(x) = \mathop{\rm {det}}(2 \pi \Sigma)^{ -\frac{1}{2} }
\exp\{-\frac{1}{2} (x-\mu)^\top \Sigma^{-1} (x-\mu) \}$](mvahtmlimg1813.gif)
with
![$\Sigma = \begin{pmatrix}1 & 0.6 \\ 0.6 & 1 \end{pmatrix}$](mvahtmlimg1814.gif)
and
![$\mu = \begin{pmatrix}0 \\ 0 \end{pmatrix}$](mvahtmlimg1815.gif)
Notice that the density is constant on ellipses. This is the reason for calling
this family of distributions ``elliptical''.