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 (in comparison with the normal distribution). Some statistical methods that were originally motivated by the study of the normal distribution have good performance for general elliptical distributions (with finite variance), particularly for spherical distributions (which are defined below). 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. 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 nonnegative-definite matrix ${\displaystyle \Sigma }$ and some scalar function ${\displaystyle \psi }$.^[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. An elliptical distribution with a density function f has the form:

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

where ${\displaystyle k}$ is the normalizing constant, ${\displaystyle x}$ is an ${\displaystyle n}$-dimensional random vector with median vector ${\displaystyle \mu }$ (which is also the mean vector if the latter exists), and ${\displaystyle \Sigma }$ is a positive definite matrix which is proportional to the covariance matrix if the latter exists.^[4]

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 mean, such as the Cauchy distribution (even in the univariate case). Because the variable x enters the density function quadratically, all elliptical distributions are symmetric about ${\displaystyle \mu .}$

If two subsets of a jointly elliptical random vector are uncorrelated, then if their means exist they are mean independent of each other (the mean of each subvector conditional on the value of the other subvector equals the unconditional mean).^{[8]: p. 748}

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 (though not necessarily with the same elliptical distribution), and any subset of X is elliptical.^{[8]: p. 748}

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.^[9][10]