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^[1]

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.^[2]

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.^[3]

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 .^[4]

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

Applications

Elliptical distributions are important in portfolio 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 (Chamberlain 1983; Owen and Rabinovitch 1983). For multi-normal distributions, location and scale correspond to mean and standard deviation.

References

↑ Stamatis Cambanis, Steel Huang, and Gordon Simons (1981). "On the Theory of Elliptically Contoured Distributions". Journal of Multivariate Analysis 11: 368–385. doi:10.1016/0047-259x(81)90082-8.
↑ Härdle and Simar (2012), p. 178.
↑ Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.
↑ Z. Landsman, E. Valdez, Tail conditional expectations for elliptical distribution North Am. Actuarial J., 7 (4) (2003), pp. 55–71

Wolfgang Karl. Härdle and Lèopold Simar (2012). Applied Multivariate Statistical Analysis (3rd ed.). Springer.
Fang, K. Kotz, S. and Ng., K. (1990). Symmetric Multivariate and Related Distributions. London: Chapman & Hall.
McNeil, Alexander; Frey, Rüdiger; Embrechts, Paul (2005). Quantitative Risk Management. Princeton University Press. ISBN 0-691-12255-5.
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
Landsman, Zinoviy M.; Valdez, Emiliano A. (2003) Tail Conditional Expectations for Elliptical Distributions (with discussion), The North American Actuarial Journal, 7, 55–123.
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

Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory-Schulz Fréchet Gamma Gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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