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.

Definition

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]

Examples

Examples include the following multivariate probability distributions:

• Multivariate normal distribution
• Multivariate t-distribution
• Symmetric multivariate stable distribution^[5]
• Symmetric multivariate Laplace distribution^[6]
• Multivariate logistic distribution^[7]
• Multivariate symmetric general hyperbolic distribution^[7]

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 ${\displaystyle f(x)}$ ) 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}

Applications

Elliptical distributions are used in statistics and in economics.

In mathematical economics, elliptical distributions have been used to describe portfolios in mathematical finance.^[9][10]

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.^[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 ${\displaystyle \alpha I}$  where ${\displaystyle I}$  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}

Notes

1. Cambanis, Huang & Simons (1981, p. 368)
2. Fang, Kotz & Ng (1990, Chapter 2.9 "Complex elliptically symmetric distributions", pp. 64-66)
3. 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).
4. Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: Applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.
5. Nolan, John (September 29, 2014). "Multivariate stable densities and distribution functions: general and elliptical case". Retrieved 2017-05-26.
6. Pascal, F.; et al. (2013). "Parameter Estimation For Multivariate Generalized Gaussian Distributions". IEEE Transactions on Signal Processing. 61 (23): 5960–5971. arXiv:1302.6498. Bibcode:2013ITSP...61.5960P. doi:10.1109/TSP.2013.2282909. S2CID 3909632.
7. Schmidt, 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. ISBN 9783642593659.
8. Owen & Rabinovitch (1983)
9. (Gupta, Varga & Bodnar 2013)
10. (Chamberlain 1983; Owen and Rabinovitch 1983)
11. 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))
12. Fang & Zhang (1990)
13. Fang & Zhang (1990, Chapter 2.8 "Distribution of quadratic forms and Cochran's theorem", pp. 74-81)
14. Fang & Zhang (1990, Chapter 2.5 "Spherical distributions", pp. 53-64)
15. Fang & Zhang (1990, Chapter IV "Estimation of parameters", pp. 127-153)
16. Fang & Zhang (1990, Chapter V "Testing hypotheses", pp. 154-187)
17. Fang & Zhang (1990, Chapter VII "Linear models", pp. 188-211)
18. Pan & Fang (2007, p. ii)
19. Kollo & von Rosen (2005, p. xiii)
20. Kariya, Takeaki; Sinha, Bimal K. (1989). Robustness of statistical tests. Academic Press. ISBN 0123982308.
21. Kollo & von Rosen (2005, p. 221)
22. Chamberlain (1983)

References

• Anderson, T. W. (2004). An introduction to multivariate statistical analysis (3rd ed.). New York: John Wiley and Sons. ISBN 9789812530967.
• Cambanis, Stamatis; Huang, Steel; Simons, Gordon (1981). "On the theory of elliptically contoured distributions". Journal of Multivariate Analysis. 11 (3): 368–385. doi:10.1016/0047-259x(81)90082-8.
• 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; Zhang, Yao-Ting (1990). Generalized multivariate analysis. Science Press (Beijing) and Springer-Verlag (Berlin). ISBN 3540176519. OCLC 622932253.
• 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. ISBN 0-412-314-304. OCLC 123206055.
• 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.

  Originally Gupta, Arjun K.; Varga, Tamas (1993). Elliptically contoured models in statistics. Mathematics and Its Applications (1st ed.). Dordrecht: Kluwer Academic Publishers. ISBN 0792326083.

• Kollo, Tõnu; von Rosen, Dietrich (2005). Advanced multivariate statistics with matrices. Dordrecht: Springer. ISBN 978-1-4020-3418-3.
• 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. JSTOR 2328079.
• Pan, Jianxin; Fang, Kaitai (2007). Growth curve models and statistical diagnostics (PDF). Springer series in statistics. Science Press (Beijing) and Springer-Verlag (New York). doi:10.1007/978-0-387-21812-0. ISBN 9780387950532. OCLC 44162563.

Further reading

• Fang, Kai-Tai; Anderson, T. W., eds. (1990). Statistical inference in elliptically contoured and related distributions. New York: Allerton Press. ISBN 0898640482. OCLC 20490516. A collection of papers.