Inprobability andstatistics, anelliptical distribution is any member of a broad family ofprobability distributions that generalize themultivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms anellipse and anellipsoid, respectively, in iso-density plots.

Instatistics, the normal distribution is used inclassicalmultivariate analysis, while elliptical distributions are used ingeneralized multivariate analysis, for the study of symmetric distributions with tails that areheavy, like themultivariate 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 inrobust statistics to evaluate proposed multivariate-statistical procedures.

Elliptical distributions are defined in terms of thecharacteristic function of probability theory. A random vector${\displaystyle X}$ on aEuclidean space has anelliptical distribution if its characteristic function${\displaystyle \varphi }$ satisfies the followingfunctional equation (for every column-vector${\displaystyle t}$)

${\displaystyle {\varphi }_{X-\mu }(t)=\psi (t^{\prime }\mathrm{\Sigma }t)}$

for somelocation parameter${\displaystyle \mu }$, somenonnegative-definite matrix${\displaystyle \mathrm{\Sigma }}$ and some scalar function${\displaystyle \psi }$.^[1] The definition of elliptical distributions forreal random-vectors has been extended to accommodate random vectors in Euclidean spaces over thefield ofcomplex numbers, so facilitating applications intime-series analysis.^[2] Computational methods are available for generatingpseudo-random vectors from elliptical distributions, for use inMonte Carlosimulations for example.^[3]

Some elliptical distributions are alternatively defined in terms of theirdensity functions. An elliptical distribution with a density functionf has the form:

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

where${\displaystyle k}$ is thenormalizing constant,${\displaystyle x}$ is an${\displaystyle n}$-dimensionalrandom vector withmedian vector${\displaystyle \mu }$ (which is also the mean vector if the latter exists), and${\displaystyle \mathrm{\Sigma }}$ is apositive definite matrix which is proportional to thecovariance matrix if the latter exists.^[4]

Examples include the following multivariate probability distributions:

• Multivariate normal distribution
• Multivariatet-distribution
• Symmetric multivariate stable distribution^[5]
• Symmetric multivariate Laplace distribution^[6]
• Multivariate logistic distribution^[7]
• Multivariate symmetric generalhyperbolic distribution^[7]

In the 2-dimensional case, if the density exists, each iso-density locus (the set ofx₁,x₂ pairs all giving a particular value of${\displaystyle f(x)}$) is anellipse or a union of ellipses (hence the name elliptical distribution). More generally, for arbitraryn, the iso-density loci are unions ofellipsoids. All these ellipsoids or ellipses have the common center μ and are scaled copies (homothets) of each other.

Themultivariate 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 undefinedmean, such as theCauchy distribution (even in the univariate case). Because the variablex enters the density function quadratically, all elliptical distributions aresymmetric about${\displaystyle \mu .}$

If two subsets of a jointly elliptical random vector areuncorrelated, then if their means exist they aremean 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 vectorX is elliptically distributed, then so isDX for any matrixD with fullrow rank. Thus any linear combination of the components ofX is elliptical (though not necessarily with the same elliptical distribution), and any subset ofX is elliptical.^{[8]: p. 748}

Elliptical distributions are used in statistics and in economics. They are also used to calculate thelanding footprints of spacecraft.

In mathematical economics, elliptical distributions have been used to describeportfolios inmathematical finance.^[9][10]

In statistics, themultivariatenormal distribution (of Gauss) is used inclassicalmultivariate 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.^[11][12] Under finite-variance assumptions, an extension ofCochran's theorem (on the distribution of quadratic forms) holds.^[13]

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

Another use of elliptical distributions is inrobust 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 thelimiting theory of statistics ("asymptotics").^[21]

Elliptical distributions are important inportfolio 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.^[22][8] Various features of portfolio analysis, includingmutual fund separation theorems and theCapital Asset Pricing Model, hold for all elliptical distributions.^{[8]: p. 748}

• Types of probability distributions
• Location-scale family probability distributions
• Multivariate statistics
• Normal distribution
• Ellipsoids

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