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

Elliptical distributions are defined in terms of the characteristic function of probability theory. A random vector ${\displaystyle X}$ on a Euclidean space has an elliptical distribution if its characteristic function ${\displaystyle \phi }$ satisfies the following functional equation (for every column-vector ${\displaystyle t}$)

${\displaystyle \phi _{X-\mu }(t)=\psi (t'\Sigma t)}$

for some location parameter ${\displaystyle \mu }$, some

${\displaystyle \Sigma }$

${\displaystyle \psi }$

where ${\displaystyle k}$ is the normalizing constant, ${\displaystyle x}$ is an ${\displaystyle n}$-dimensional

${\displaystyle \mu }$ (which is also the mean vector if the latter exists), and ${\displaystyle \Sigma }$ is a

• Multivariate normal distribution
• Multivariate t-distribution
• Symmetric multivariate stable distribution^[5]
• Symmetric multivariate Laplace distribution^[6]
• Multivariate logistic distribution^[7]
• Multivariate symmetric general hyperbolic distribution^[7]

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

There exist elliptical distributions that have undefined

${\displaystyle \mu .}$

If two subsets of a jointly elliptical random vector are

For suitable elliptical distributions, some classical methods continue to have good properties.[11]^[12] Under finite-variance assumptions, an extension of Cochran's theorem (on the distribution of quadratic forms) holds.^[13]

Spherical distribution

An elliptical distribution with a zero mean and variance in the form ${\displaystyle \alpha I}$ where ${\displaystyle I}$ is the identity-matrix is called a spherical distribution.^[14] For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended.^[15][16] Similar results hold for linear models,^[17] and indeed also for complicated models (especially for the growth curve model). The analysis of multivariate models uses multilinear algebra (particularly Kronecker products and vectorization) and matrix calculus.^[12][18][19]

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,^[20] for example by using the limiting theory of statistics ("asymptotics").^[21]

Economics and finance

Elliptical distributions are important in