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.
In statistics, the normal distribution is used in classical multivariate analysis, while elliptical distributions are used in generalized multivariate analysis, for the study of symmetric distributions with tails that are heavy, like the multivariate t-distribution, or light. Some statistical methods that were originally motivated by the study of the normal distribution have good performance for general elliptical distributions, particularly for spherical distributions. Elliptical distributions are also used in robust statistics to evaluate proposed multivariate-statistical procedures.

Definition

Elliptical distributions are defined in terms of the characteristic function of probability theory. A random vector on a Euclidean space has an elliptical distribution if its characteristic function satisfies the following functional equation
for some location parameter, some nonnegative-definite matrix and some scalar function . The definition of elliptical distributions for real random-vectors has been extended to accommodate random vectors in Euclidean spaces over the field of complex numbers, so facilitating applications in time-series analysis. Computational methods are available for generating pseudo-random vectors from elliptical distributions, for use in Monte Carlo simulations for example.
Some elliptical distributions are alternatively defined in terms of their density functions. An elliptical distribution with a density function f has the form:
where is the normalizing constant, is an -dimensional random vector with median vector , and is a positive definite matrix which is proportional to the covariance matrix if the latter exists.

Examples

Examples include the following multivariate probability distributions:
In the 2-dimensional case, if the density exists, each iso-density locus is an ellipse or a union of ellipses. 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 of each other.
The multivariate normal distribution is the special case in which. While the multivariate normal is unbounded, in general elliptical distributions can be bounded or unbounded—such a distribution is bounded if for all greater than some value.
There exist elliptical distributions that have undefined mean, such as the Cauchy distribution. Because the variable x enters the density function quadratically, all elliptical distributions are symmetric about
If two subsets of a jointly elliptical random vector are uncorrelated, then if their means exist they are mean independent of each other.
If random vector X is elliptically distributed, then so is DX for any matrix D with full row rank. Thus any linear combination of the components of X is elliptical, and any subset of X is elliptical.

Applications

Elliptical distributions are used in statistics and in economics.
In mathematical economics, elliptical distributions have been used to describe portfolios in mathematical finance.

Statistics: Generalized multivariate analysis

In statistics, the multivariate normal distribution is used in classical multivariate analysis, in which most methods for estimation and hypothesis-testing are motivated for the normal distribution. In contrast to classical multivariate analysis, generalized multivariate analysis refers to research on elliptical distributions without the restriction of normality.
For suitable elliptical distributions, some classical methods continue to have good properties. Under finite-variance assumptions, an extension of Cochran's theorem holds.

Spherical distribution

An elliptical distribution with a zero mean and variance in the form where is the identity-matrix is called a spherical distribution. For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended. Similar results hold for linear models, and indeed also for complicated models. The analysis of multivariate models uses multilinear algebra and matrix calculus.

Robust statistics: Asymptotics

Another use of elliptical distributions is in robust statistics, in which researchers examine how statistical procedures perform on the class of elliptical distributions, to gain insight into the procedures' performance on even more general problems, for example by using the limiting theory of statistics.

Economics and finance

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. Various features of portfolio analysis, including mutual fund separation theorems and the Capital Asset Pricing Model, hold for all elliptical distributions.