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Difference between revisions of "Degenerate distribution"

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''in an $n$-dimensional Euclidean space''
 
''in an $n$-dimensional Euclidean space''
  
Any [[Probability distribution|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.
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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.
  
  
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====Comments====
 
====Comments====
 
An improper distribution also refers, in Bayesian statistics (cf. [[Bayesian approach|Bayesian approach]]; [[Bayesian approach, empirical|Bayesian approach, empirical]]), to a measure of infinite total mass, which is still being manipulated as a probability distribution. Cf. also [[Improper distribution|Improper distribution]].
 
An improper distribution also refers, in Bayesian statistics (cf. [[Bayesian approach|Bayesian approach]]; [[Bayesian approach, empirical|Bayesian approach, empirical]]), to a measure of infinite total mass, which is still being manipulated as a probability distribution. Cf. also [[Improper distribution|Improper distribution]].
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[[Category:Probability and statistics]]

Latest revision as of 19:24, 26 October 2014

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://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