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| {{Expert-subject|Statistics|date=November 2008}}
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| In [[statistics]], '''Wilks' lambda distribution''' (named for [[Samuel S. Wilks]]), is a [[probability distribution]] used in multivariate hypothesis testing, especially with regard to the [[likelihood-ratio test]] and [[Multivariate analysis of variance]]. It is a multivariate generalization of the univariate [[F-distribution]], and generalizes the [[F-distribution]] in the same way that the [[Hotelling's T-squared distribution]] generalizes [[Student's t-distribution]].
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| Wilks' lambda distribution is related to two [[statistical independence|independent]] [[Wishart distribution|Wishart distributed]] variables, and is
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| defined as follows,<ref name="MKB">
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| {{cite book
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| | last = Mardia
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| | first = K.V.
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| | coauthors = J.T. Kent, J.M. Bibby
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| | title = Multivariate Analysis
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| | publisher = Academic Press
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| | year = 1979
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| }}
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| </ref>
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| given
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| :<math>A \sim W_p(\Sigma, m) \qquad B \sim W_p(\Sigma, n)</math> | |
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| independent and with <math>m \ge p</math>
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| :<math>\lambda = \frac{\det(A)}{\det(A+B)} = \frac{1}{\det(I+A^{-1}B)} \sim \Lambda(p,m,n)</math>
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| where ''p'' is the number of dimensions. In the context of [[likelihood-ratio test]]s ''m'' is typically the error degrees of freedom, and ''n'' is the hypothesis degrees of freedom, so that <math>n+m</math> is the total degrees of freedom.<ref name="MKB"/>
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| The distribution can be related to a product of [[statistical independence|independent]] [[Beta distribution|Beta distributed]] random variables
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| :<math>u_i \sim B\left(\frac{m+i-p}{2},\frac{p}{2}\right)</math> | |
| :<math>\prod_{i=1}^n u_i \sim \Lambda(p,m,n).</math>
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| For large ''m'' Bartlett's approximation
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| <ref>
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| {{cite journal
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| | last = Bartlett
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| | first = M.S.
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| | title = A note on multiplying factors for various <math>\chi^2</math> approximations
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| | journal = Journal of the Royal Statistical Society, Series B
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| | volume = 16
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| | pages = 296–298
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| | year = 1954
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| }}
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| </ref>
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| allows Wilks' lambda to be approximated with a [[Chi-squared distribution]]
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| :<math>\left(\frac{p+n+1}{2}-m\right)\log \Lambda(p,m,n) \sim \chi^2_{np}.</math><ref name="MKB"/>
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| == See also ==
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| {{Colbegin}}
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| * [[Chi-squared distribution]]
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| * [[F-distribution]]
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| * [[Gamma distribution]]
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| * [[Hotelling's T-squared distribution]]
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| * [[Student's t-distribution]]
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| * [[Wishart distribution]]
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| {{Colend}}
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| ==References==
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| <references/>
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| {{ProbDistributions|continuous-semi-infinite}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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