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.

Definition

Elliptical distributions can be defined using characteristic functions. A multivariate distribution is said to be elliptical if its characteristic function is of the form

$e^{it'\mu }\Psi (t'\Sigma t)\,$

for a specified vector $\mu$, positive-definite matrix $\Sigma$, and characteristic function $\Psi$. The function $\Psi$ is known as the characteristic generator of the elliptical distribution.

Elliptical distributions can also be defined in terms of their density functions. When they exist, the density functions f have the structure:

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

where $k$ is the scale factor, $x$ is an $n$-dimensional random vector with median vector $\mu$ (which is also the mean vector if the latter exists), $\Sigma$ is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and $g$ is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.

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

Note that there exist elliptical distributions that have infinite mean and variance, such as the multivariate Student's t-distribution or the multivariate Cauchy distribution .

Because the variable x enters the density function quadratically, all elliptical distributions are symmetric about $\mu .$

Applications

Elliptical distributions are used in statistics and in economics.

In mathematical economics, elliptical distributions have been used to describe portfolios in mathematical finance.

Statistics: Generalized multivariate analysis

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. 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 (on the distribution of quadratic forms) holds.

Spherical distribution

An elliptical distribution with a zero mean and identity-matrix variance is called a spherical distribution. For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended. Similar results hold for linear models, 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.

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, for example by using the limiting theory of statistics ("asymptotics").

Citations

References

• Anderson, T. W. (2004). An introduction to multivariate statistical analysis (3rd ed.). New York: John Wiley and Sons. ISBN 9789812530967. {{cite book}}: Invalid |ref=harv (help)
• Chamberlain, G. (1983). "A characterization of the distributions that imply mean-variance utility functions", Journal of Economic Theory 29, 185–201. doi:10.1016/0022-0531(83)90129-1
• Fang, Kai-Tai; Zhang, Yao-Ting (1990). Generalized multivariate analysis. Science Press (Beijing) and Springer-Verlag (Berlin). ISBN 3540176519. 9783540176510. {{cite book}}: Invalid |ref=harv (help)
• Fang, Kai-Tai; Kotz, Samuel; Ng, Kai-Wang (1990). Symmetric multivariate and related distributions. London: Chapman & Hall. {{cite book}}: Invalid |ref=harv (help)
• Gupta, Arjun K.; Varga, Tamas; Bodnar, Taras (2013). Elliptically contoured models in statistics and portfolio theory (2nd ed.). New York: Springer-Verlag. doi:10.1007/978-1-4614-8154-6. ISBN 978-1-4614-8153-9. {{cite book}}: Invalid |ref=harv (help)
• Härdle, Wolfgang Karl; Simar, Lèopold (2012). Applied multivariate statistical analysis (3rd ed.). Springer. {{cite book}}: Invalid |ref=harv (help)
• Kollo, Tõnu; von Rosen, Dietrich (2005). Advanced multivariate statistics with matrices. Dordrecht: Springer. ISBN 978-1-4020-3418-3. {{cite book}}: Invalid |ref=harv (help)
• Landsman, Zinoviy M.; Valdez, Emiliano A. (2003) Tail Conditional Expectations for Elliptical Distributions (with discussion), The North American Actuarial Journal, 7, 55–123.
• McNeil, Alexander; Frey, Rüdiger; Embrechts, Paul (2005). Quantitative Risk Management. Princeton University Press. ISBN 0-691-12255-5.
• Owen, J., and Rabinovitch, R. (1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice", Journal of Finance 38, 745–752. JSTOR 2328079
• Pan, Jianxin; Fang, Kaitai (2007). Growth curve models and statistical diagnostics. Beijing: Science Press. ISBN 9780387950532. {{cite book}}: Invalid |ref=harv (help)