Elliptical distribution

Elliptical Distribution

Elliptical distributions are a broad class of probability distributions that generalize the multivariate normal distribution. They are important in the field of statistics and have applications in finance, economics, and various branches of science. This article provides an overview of elliptical distributions, their properties, and their applications.

Definition

An elliptical distribution is a type of probability distribution for a random vector \( \mathbf{X} \in \mathbb{R}^n \) that can be expressed in the form:

\[ \mathbf{X} = \boldsymbol{\mu} + R \mathbf{A} \mathbf{U}, \]

where:

• \( \boldsymbol{\mu} \) is a location vector in \( \mathbb{R}^n \),
• \( R \) is a non-negative random variable,
• \( \mathbf{A} \) is a deterministic \( n \times n \) matrix,
• \( \mathbf{U} \) is a random vector uniformly distributed on the unit sphere in \( \mathbb{R}^n \).

The distribution of \( \mathbf{X} \) is said to be elliptical if its characteristic function has the form:

\[ \varphi_{\mathbf{X}}(\mathbf{t}) = \exp\left(i \mathbf{t}' \boldsymbol{\mu} \right) \psi(\mathbf{t}' \Sigma \mathbf{t}), \]

where \( \Sigma = \mathbf{A} \mathbf{A}' \) is a positive semi-definite matrix and \( \psi \) is a characteristic generator function.

Properties

• Symmetry: Elliptical distributions are symmetric about their location vector \( \boldsymbol{\mu} \).
• Contours: The contours of the density function of an elliptical distribution are ellipsoids centered at \( \boldsymbol{\mu} \).
• Marginals: Any linear combination of components of an elliptical distribution is also elliptically distributed.
• Tail Behavior: The tail behavior of an elliptical distribution is determined by the generator function \( \psi \).

Examples

• Multivariate Normal Distribution: The most well-known elliptical distribution, where \( R \) follows a chi distribution.
• Multivariate t-Distribution: An elliptical distribution with heavier tails than the normal distribution.
• Multivariate Cauchy Distribution: A special case of the multivariate t-distribution with one degree of freedom.

Applications

Elliptical distributions are used in various fields due to their flexibility and ability to model data with elliptical contours.

Finance

In finance, elliptical distributions are used to model the joint distribution of asset returns. The multivariate normal distribution is often used in portfolio theory, while the multivariate t-distribution is used to account for heavy tails in financial returns.

Signal Processing

In signal processing, elliptical distributions are used to model noise and other random processes. The symmetry and contour properties make them suitable for various estimation and detection problems.

See Also

• Multivariate normal distribution
• Probability distribution
• Covariance matrix

References

• Fang, K.-T., Kotz, S., & Ng, K. W. (1990). "Symmetric Multivariate and Related Distributions." Chapman and Hall.
• Cambanis, S., Huang, S., & Simons, G. (1981). "On the theory of elliptically contoured distributions." Journal of Multivariate Analysis, 11(3), 368-385.