Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid, respectively, in iso-density plots.
Elliptical distributions are also used in robust statistics to evaluate proposed multivariate-statistical procedures.
Elliptical distributions are defined in terms of the characteristic function of probability theory.
on a Euclidean space has an elliptical distribution if its characteristic function
[1] 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.
[2] Computational methods are available for generating pseudo-random vectors from elliptical distributions, for use in Monte Carlo simulations for example.
[3] Some elliptical distributions are alternatively defined in terms of their density functions.
[4] Examples include the following multivariate probability distributions: 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
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.
can take on arbitrarily large positive or negative values with non-zero probability, because
Because the variable x enters the density function quadratically, all elliptical distributions are symmetric about
748 If random vector X is elliptically distributed, then so is DX for any matrix D with full row rank.
In statistics, the multivariate normal distribution (of Gauss) is used in classical multivariate analysis, in which most methods for estimation and hypothesis-testing are motivated for the normal distribution.
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.
[14] For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended.
The analysis of multivariate models uses multilinear algebra (particularly Kronecker products and vectorization) and matrix calculus.
[12][18][19] 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").
[22][8] Various features of portfolio analysis, including mutual fund separation theorems and the Capital Asset Pricing Model, hold for all elliptical distributions.[8]: p.