Difference between revisions of "Degenerate distribution"
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| − | Any [[ | + | 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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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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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.
Degenerate distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degenerate_distribution&oldid=15645