Elliptical distribution

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

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

$e^{it'\mu} \Psi(t' \Sigma t) \,$

for a specified vector $\mu$, positive-definite matrix $\Sigma$, and characteristic function $\Psi$. The function $\Psi$ is known as the characteristic generator of the elliptical distribution.

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

$f(x)= k \cdot g((x-\mu)'\Sigma^{-1}(x-\mu))$

where $k$ is the scale factor, $x$ is an $n$-dimensional random vector with median vector $\mu$ (which is also the mean vector if the latter exists), $\Sigma$ is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and $g$ is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.

Properties

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 $f(x)$) 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 $g(z)=e^{-z/2}$. While the multivariate normal is unbounded (each element of $x$ can take on arbitrarily large positive or negative values with non-zero probability, because $e^{-z/2}>0$ for all non-negative $z$), in general elliptical distributions can be bounded or unbounded—such a distribution is bounded if $g(z)=0$ for all $z$ 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 .

Because the variable x enters the density function quadratically, all elliptical distributions are symmetric about $\mu.$

Applications

Elliptical distributions are important in portfolio theory because, if the returns on all assets available for portfolio formation are jointly elliptically distributed, then all portfolios can be characterized completely by their location and scale – that is, any two portfolios with identical location and scale of portfolio return have identical distributions of portfolio return (Chamberlain 1983; Owen and Rabinovitch 1983). For multi-normal distributions, location and scale correspond to mean and standard deviation.

References

• 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
• Landsman, Zinoviy M.; Valdez, Emiliano A. (2003) Tail Conditional Expectations for Elliptical Distributions (with discussion), The North American Actuarial Journal, 7, 55–123.
• 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