Elliptical distribution

{{Short description|Family of distributions that generalize the multivariate normal distribution}}

In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. 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 $X$ on a Euclidean space has an elliptical distribution if its characteristic function $\phi$ satisfies the following functional equation (for every column-vector $t$ )

: $\phi_{X-\mu}(t)
=
\psi(t' \Sigma t)$

for some location parameter $\mu$ , some nonnegative-definite matrix $\Sigma$ and some scalar function $\psi$ .{{harvtxt|Cambanis|Huang|Simons|1981|p=368}} 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.{{harvtxt|Fang|Kotz|Ng|1990|loc=Chapter 2.9 "Complex elliptically symmetric distributions", pp. 64-66}} Computational methods are available for generating pseudo-random vectors from elliptical distributions, for use in Monte Carlo simulations for example.{{harvtxt|Johnson|1987|loc=Chapter 6, "Elliptically contoured distributions, pp. 106-124}}: {{cite book|last=Johnson|first=Mark E.|title=Multivariate statistical simulation: A guide to selecting and generating continuous multivariate distributions|publisher=John Wiley and Sons|year=1987}}, "an admirably lucid discussion" according to {{harvtxt|Fang|Kotz|Ng|1990|p=27}}.

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

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

where $k$ is the normalizing constant, $x$ is an $n$ -dimensional random vector with median vector $\mu$ (which is also the mean vector if the latter exists), and $\Sigma$ is a positive definite matrix which is proportional to the covariance matrix if the latter exists.Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: Applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.

=Examples=

Examples include the following multivariate probability distributions:

Multivariate normal distribution
Multivariate t-distribution
Symmetric multivariate stable distribution{{cite web|title=Multivariate stable densities and distribution functions: general and elliptical case|author=Nolan, John|url=https://www.researchgate.net/publication/246910601|access-date=2017-05-26|date=September 29, 2014}}
Symmetric multivariate Laplace distribution{{cite journal|title=Parameter Estimation For Multivariate Generalized Gaussian Distributions|journal=IEEE Transactions on Signal Processing|volume=61|issue=23|pages=5960–5971|author=Pascal, F.|display-authors=etal|arxiv=1302.6498|doi=10.1109/TSP.2013.2282909|year=2013|bibcode=2013ITSP...61.5960P |s2cid=3909632}}
Multivariate logistic distribution{{cite book|title=Credit Risk: Measurement, Evaluation and Management|page=274|author=Schmidt, Rafael|chapter=Credit Risk Modeling and Estimation via Elliptical Copulae|editor=Bol, George|display-editors=etal|year=2012|publisher=Springer|isbn=9783642593659}}
Multivariate symmetric general hyperbolic distribution

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.

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 $\mu.$

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).{{harvtxt|Owen|Rabinovitch|1983}}{{rp|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.{{rp|p. 748}}

Applications

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

In mathematical economics, elliptical distributions have been used to describe portfolios in mathematical finance.{{harv|Gupta|Varga|Bodnar|2013}}(Chamberlain 1983; Owen and Rabinovitch 1983)

=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.{{harvtxt|Anderson|2004|loc=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)}}{{harvtxt|Fang|Zhang|1990}} Under finite-variance assumptions, an extension of Cochran's theorem (on the distribution of quadratic forms) holds.{{harvtxt|Fang|Zhang|1990|loc=Chapter 2.8 "Distribution of quadratic forms and Cochran's theorem", pp. 74-81}}

==Spherical distribution==

An elliptical distribution with a zero mean and variance in the form $\alpha I$ where $I$ is the identity-matrix is called a spherical distribution.{{harvtxt|Fang|Zhang|1990|loc=Chapter 2.5 "Spherical distributions", pp. 53-64}} For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended.{{harvtxt|Fang|Zhang|1990|loc=Chapter IV "Estimation of parameters", pp. 127-153}}{{harvtxt|Fang|Zhang|1990|loc=Chapter V "Testing hypotheses", pp. 154-187}} Similar results hold for linear models,{{harvtxt|Fang|Zhang|1990|loc=Chapter VII "Linear models", pp. 188-211}} 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.{{harvtxt|Pan|Fang|2007|p=ii}}{{harvtxt|Kollo|von Rosen|2005|p=xiii}}

==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,{{cite book|last1=Kariya|first1=Takeaki|first2=Bimal K.|last2=Sinha|title=Robustness of statistical tests|publisher=Academic Press|year=1989|isbn=0123982308}}

for example by using the limiting theory of statistics ("asymptotics").{{harvtxt|Kollo|von Rosen|2005|p=221}}

=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.{{harvtxt|Chamberlain|1983}} Various features of portfolio analysis, including mutual fund separation theorems and the Capital Asset Pricing Model, hold for all elliptical distributions.{{rp|p. 748}}

Notes

References

{{cite book | last = Anderson | first = T. W. |author-link=Theodore W. Anderson| title = An introduction to multivariate statistical analysis | publisher = John Wiley and Sons | location = New York | year = 2004 | edition = 3rd | isbn = 9789812530967}}
{{cite journal|first1=Stamatis|last1=Cambanis|first2=Steel|last2=Huang |first3=Gordon|last3=Simons

| title=On the theory of elliptically contoured distributions

|journal= Journal of Multivariate Analysis| volume=11|issue=3| year=1981| pages=368–385| doi=10.1016/0047-259x(81)90082-8|doi-access=free}}

{{cite journal |last1=Chamberlain |first1=Gary |title=A characterization of the distributions that imply mean—Variance utility functions |journal=Journal of Economic Theory |date=February 1983 |volume=29 |issue=1 |pages=185–201 |doi=10.1016/0022-0531(83)90129-1}}
{{cite book|title=Generalized multivariate analysis

|oclc=622932253 }}

{{cite book |title=Symmetric multivariate and related distributions |last1=Fang|first1=Kai-Tai|author-link1=Kai-Tai Fang|last2=Kotz|first2=Samuel|author-link2=Samuel Kotz|last3=Ng|first3=Kai Wang ("Kai-Wang" on front cover)|year=1990|series=Monographs on statistics and applied probability|volume=36|publisher=Chapman and Hall|location=London|isbn=0-412-314-304|oclc=123206055}}
{{cite book|title=Elliptically contoured models in statistics and portfolio theory |first1=Arjun K.|last1=Gupta |first2=Tamas|last2=Varga |first3=Taras|last3=Bodnar

|year=2013|publisher=Springer-Verlag |location=New York |doi=10.1007/978-1-4614-8154-6 |isbn=978-1-4614-8153-9|edition=2nd}}

:Originally {{cite book|title=Elliptically contoured models in statistics|first1=Arjun K.|last1=Gupta |first2=Tamas|last2=Varga

|year=1993|publisher=Kluwer Academic Publishers |location=Dordrecht |isbn=0792326083|edition=1st|series=Mathematics and Its Applications}}

{{cite book|last1=Kollo|first1=Tõnu|last2=von Rosen|first2=Dietrich |title=Advanced multivariate statistics with matrices |location=Dordrecht |publisher=Springer |year=2005 |isbn=978-1-4020-3418-3 }}
{{cite journal |last1=Owen |first1=Joel |last2=Rabinovitch |first2=Ramon |title=On the Class of Elliptical Distributions and their Applications to the Theory of Portfolio Choice |journal=The Journal of Finance |date=June 1983 |volume=38 |issue=3 |pages=745–752 |doi=10.2307/2328079|jstor=2328079 }}
{{cite book

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|last2=Fang

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|author-link2=Kaitai Fang

|title=Growth curve models and statistical diagnostics

|publisher=Science Press (Beijing) and Springer-Verlag (New York)

|series=Springer series in statistics

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|doi=10.1007/978-0-387-21812-0

|isbn=9780387950532

|url=http://eprints.maths.manchester.ac.uk/561/1/Growth_Curve.pdf

}}