In
statistics, the normal distribution is used in classicalmultivariate 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.
Definition
Elliptical distributions are defined in terms of the
characteristic function of probability theory. A random vector on a
Euclidean space has an elliptical distribution if its characteristic function satisfies the following
functional equation (for every column-vector )
Some elliptical distributions are alternatively defined in terms of their
density functions. An elliptical distribution with a density function f has the form:
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 ) 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 . While the multivariate normal is unbounded (each element of can take on arbitrarily large positive or negative values with non-zero probability, because for all non-negative ), in general elliptical distributions can be bounded or unbounded—such a distribution is bounded if for all 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
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
Applications
Elliptical distributions are used in statistics and in economics.
In statistics, the
multivariate normal distribution (of Gauss) is used in classicalmultivariate 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 of
Cochran's theorem (on the distribution of quadratic forms) holds.[13]
Spherical distribution
An elliptical distribution with a zero mean and variance in the form where is the identity-matrix is called a spherical distribution.[14] For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended.[15][16] Similar results hold for
linear models,[17] 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.[12][18][19]
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,[20] for example by using the
limiting theory of statistics ("asymptotics").[21]
Economics and finance
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.[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. 748
^Fang, Kotz & Ng (1990, Chapter 2.9 "Complex elliptically symmetric distributions", pp. 64-66)
^Johnson (1987, Chapter 6, "Elliptically contoured distributions, pp. 106-124): Johnson, Mark E. (1987). Multivariate statistical simulation: A guide to selecting and generating continuous multivariate distributions. John Wiley and Sons., "an admirably lucid discussion" according to
Fang, Kotz & Ng (1990, p. 27).
^Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: Applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.
^
abSchmidt, Rafael (2012). "Credit Risk Modeling and Estimation via Elliptical Copulae". In Bol, George; et al. (eds.). Credit Risk: Measurement, Evaluation and Management. Springer. p. 274.
ISBN9783642593659.
^Anderson (2004, The final section of the text (before "Problems") that are always entitled "Elliptically contoured distributions", of the following chapters: Chapters 3 ("Estimation of the mean vector and the covariance matrix", Section 3.6, pp. 101-108), 4 ("The distributions and uses of sample correlation coefficients", Section 4.5, pp. 158-163), 5 ("The generalized T2-statistic", Section 5.7, pp. 199-201), 7 ("The distribution of the sample covariance matrix and the sample generalized variance", Section 7.9, pp. 242-248), 8 ("Testing the general linear hypothesis; multivariate analysis of variance", Section 8.11, pp. 370-374), 9 ("Testing independence of sets of variates", Section 9.11, pp. 404-408), 10 ("Testing hypotheses of equality of covariance matrices and equality of mean vectors and covariance vectors", Section 10.11, pp. 449-454), 11 ("Principal components", Section 11.8, pp. 482-483), 13 ("The distribution of characteristic roots and vectors", Section 13.8, pp. 563-567))
Chamberlain, Gary (February 1983). "A characterization of the distributions that imply mean—Variance utility functions". Journal of Economic Theory. 29 (1): 185–201.
doi:
10.1016/0022-0531(83)90129-1.
Fang, Kai-Tai;
Kotz, Samuel; Ng, Kai Wang ("Kai-Wang" on front cover) (1990). Symmetric multivariate and related distributions. Monographs on statistics and applied probability. Vol. 36. London: Chapman and Hall.
ISBN0-412-314-304.
OCLC123206055.
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.
ISBN978-1-4614-8153-9.
Originally Gupta, Arjun K.; Varga, Tamas (1993). Elliptically contoured models in statistics. Mathematics and Its Applications (1st ed.). Dordrecht: Kluwer Academic Publishers.
ISBN0792326083.
Kollo, Tõnu; von Rosen, Dietrich (2005). Advanced multivariate statistics with matrices. Dordrecht: Springer.
ISBN978-1-4020-3418-3.
Owen, Joel; Rabinovitch, Ramon (June 1983). "On the Class of Elliptical Distributions and their Applications to the Theory of Portfolio Choice". The Journal of Finance. 38 (3): 745–752.
doi:
10.2307/2328079.
JSTOR2328079.