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 $(p \times 1)$ random vector $Y$ is said to have a spherical distribution $S_p(\phi)$ if its characteristic function $\psi_Y(t)$ satisfies: $\psi_Y(t) = \phi(t^\top t)$ for some scalar function $\phi(.)$ which is then called the characteristic generator of the spherical distribution $S_p(\phi)$. We will write $Y \sim S_p(\phi)$.

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 $N_p(0,\data{I}_{p})$.

THEOREM 5.10   Spherical random variables have the following properties:

1. All marginal distributions of a spherical distributed random vector are spherical.
2. All the marginal characteristic functions have the same generator.

3. Let $X \sim S_p(\phi)$, then $X$ has the same distribution as $r u^{(p)}$ where $u^{(p)}$ is a random vector distributed uniformly on the unit sphere surface in $\mathbb{R}^p$ and $r \geq 0$ is a random variable independent of $u^{(p)}$. If $E (r^2) < \infty$, then
   
   $$E (X) = 0 \ , \quad Cov(X) = \frac{E (r^2)}{p}\data{I}_p.$$
   
   

The random radius $r$ is related to the generator $\phi$ by a relation described in Fang et al. (1990, p.29). The moments of $X \sim S_p(\phi)$, 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 $p-1$ are continuous and the marginal densities of dimension smaller than $p-2$ are differentiable (except possibly at the origin in both cases). Univariate marginal densities for $p$ greater than $2$ are nondecreasing on $(-\infty, 0)$ and nonincreasing on $(0, \infty)$.

DEFINITION 5.2   A $(p \times 1)$ random vector $X$ is said to have an elliptical distribution with parameters $\mu(p\times 1)$ and $\Sigma(p \times p)$ if $X$ has the same distribution as $\mu + \data{A}^\top Y$, where $Y \sim S_k(\phi)$ and $\data{A}$ is a $(k \times p)$ matrix such that $\data{A}^\top \data{A}=\Sigma$ with $\mathop{\rm {rank}}(\Sigma)=k$. We shall write $X \sim EC_p(\mu,\Sigma,\phi)$.

REMARK 5.1   The elliptical distribution can be seen as an extension of $N_p(\mu,\Sigma)$.

EXAMPLE 5.6   The multivariate t-distribution. Let $Z \sim N_p(0,\data{I}_{p})$ and $s \sim \chi^2_m$ be independent. The random vector $$Y=\sqrt{m} \ \frac{Z}{s}$$ has a multivariate $t$-distribution with $m$ degrees of freedom. Moreover the $t$-distribution belongs to the family of p-dimensioned spherical distributions.

EXAMPLE 5.7   The multinormal distribution. Let $X \sim N_p (\mu, \Sigma)$. Then $X \sim EC_p(\mu,\Sigma,\phi)$ and $\phi(u)=\exp{(-u/2)}$. Figure 4.3 shows a density surface of the multivariate normal distribution: $f(x) = \mathop{\rm {det}}(2 \pi \Sigma)^{ -\frac{1}{2} } \exp\{-\frac{1}{2} (x-\mu)^\top \Sigma^{-1} (x-\mu) \}$ with $\Sigma = \begin{pmatrix}1 & 0.6 \\ 0.6 & 1 \end{pmatrix}$ and $\mu = \begin{pmatrix}0 \\ 0 \end{pmatrix}$ Notice that the density is constant on ellipses. This is the reason for calling this family of distributions ``elliptical''.

THEOREM 5.11   Elliptical random vectors $X$ have the following properties:

1. Any linear combination of elliptically distributed variables are elliptical.

2. Marginal distributions of elliptically distributed variables are elliptical.

3. A scalar function $\phi(.)$ can determine an elliptical distribution $EC_p(\mu,\Sigma,\phi)$ for every $\mu \in \mathbb{R}^p$ and $\Sigma \geq 0$ with $\mathop{\rm {rank}}(\Sigma)=k$ iff $\phi(t^\top t)$ is a $p$-dimensional characteristic function.

4. Assume that $X$ is nondegenerate. If $X \sim EC_p(\mu,\Sigma,\phi)$ and $X \sim EC_p(\mu^*,\Sigma^*,\phi^*)$, then there exists a constant $c > 0$ such that
   
   $$\mu=\mu^*, \quad \Sigma=c\Sigma^*, \quad \phi^*(.) = \phi(c^{-1}.).$$
   
   
   In other words $\Sigma, \phi, \data{A}$ are not unique, unless we impose the condition that $\mathop{\rm {det}}(\Sigma)=1$.

5. The characteristic function of $X, \psi(t) = E (e^{\textrm{\bf i}t^\top X})$ is of the form
   
   $$\psi(t) = e^{\textrm{\bf i}t^\top \mu}\phi(t^\top \Sigma t)$$
   
   
   for a scalar function $\phi$.
   

6. $X \sim EC_p(\mu,\Sigma,\phi)$ with $\mathop{\rm {rank}}(\Sigma)=k$ iff $X$ has the same distribution as:
   $$\begin{gather} \mu + r \data{A}^\top u^{(k)} \end{gather}$$
   where $r \geq 0$ is independent of $u^{(k)}$ which is a random vector distributed uniformly on the unit sphere surface in $\mathbb{R}^k$ and $\data{A}$ is a $(k \times p)$ matrix such that $\data{A}^\top \data{A}=\Sigma$.

7. Assume that $X \sim EC_p(\mu,\Sigma,\phi)$ and $E (r^2) < \infty$. Then
   
   $$E (X) = \mu \quad Cov(X) = \frac{E (r^2)}{\mathop{\rm {rank}}(\Sigma)}\Sigma= - 2\phi^\top (0)\Sigma.$$
   
   

8. Assume that $X \sim EC_p(\mu,\Sigma,\phi)$ with $\mathop{\rm {rank}}(\Sigma)=k$. Then
   
   $$Q(X)= (X - \mu)^\top \Sigma^-(X - \mu)$$
   
   
   has the same distribution as $r^2$ in equation (5.18).