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Degenerate distribution

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in an $n$-dimensional Euclidean space

Any probability distribution having support on some (linear) manifold of dimension smaller than $n$. Otherwise the distribution is called non-degenerate. A degenerate distribution in the case of finite second moments is characterized by the fact that the rank $r$ of the corresponding covariance (or correlation) matrix is smaller than $n$. Here $r$ coincides with the smallest dimension of the linear manifolds on which the given degenerate distribution is supported. The concept of a degenerate distribution can be clearly extended to distributions in linear spaces. The name improper distributions is sometimes given to degenerate distributions, while non-degenerate distributions are sometimes called proper distributions.


Comments

An improper distribution also refers, in Bayesian statistics (cf. Bayesian approach; Bayesian approach, empirical), to a measure of infinite total mass, which is still being manipulated as a probability distribution. Cf. also Improper distribution.

How to Cite This Entry:
Degenerate distribution. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Degenerate_distribution&oldid=34073
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article