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 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. Elliptical distributions are defined in terms of the characteristic function of probability theory. A random vector X {displaystyle X} on a Euclidean space has an elliptical distribution if its characteristic function ϕ {displaystyle phi } satisfies the following functional equation (for every column-vector t {displaystyle t} ) for some location parameter μ {displaystyle mu } , some nonnegative-definite matrix Σ {displaystyle Sigma } and some scalar function ψ {displaystyle psi } . 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. Computational methods are available for generating pseudo-random vectors from elliptical distributions, for use in Monte Carlo simulations for example. Some elliptical distributions are alternatively defined in terms of their density functions. An elliptical distribution with a density function f has the form: where k {displaystyle k} is the normalizing constant, x {displaystyle x} is an n {displaystyle n} -dimensional random vector with median vector μ {displaystyle mu } (which is also the mean vector if the latter exists), and Σ {displaystyle Sigma } is a positive definite matrix which is proportional to the covariance matrix if the latter exists.