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. 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 equationfor 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:- Multivariate normal distribution
- Multivariate t-distribution
- Symmetric multivariate stable distribution
- Symmetric multivariate Laplace distribution
- Multivariate logistic distribution
- Multivariate symmetric general hyperbolic distribution
Properties
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. They are also used to calculate the landing footprints of spacecraft.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.