Elliptical distributions
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An elliptical distribution is a multivariate distribution
whose (multivariate) characteristic function, , being a vector with
the same number of entries as there are dimensions for the distribution, of the
form:
where is a specified
variable, and is a positive definite
matrix.
If the distribution has a probability density function then
it will take the form:
where is a scale factor, is an n-dimensional
random vector with median vector (which is also the
mean vector, if the latter exists), is a positive definite matrix which is
proportional to the covariance matrix if the latter exists, and is a
function mapping non-negative real numbers to non-negative real numbers with
finite area under the curve.
Perhaps the best known elliptical distributions are
multivariate normal (i.e. Gaussian) distributions.