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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.

 


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