# Wilks' lambda distribution

Wilks' lambda distribution

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. It is a generalization of the F-distribution, and generalizes Hotelling's T-square distribution in the same way that the F-distribution generalizes Student's t-distribution.

Wilks' lambda distribution is related to two independent Wishart distributed variables, and isdefined as follows,cite book
last = Mardia
first = K.V.
coauthors = J.T. Kent, J.M. Bibby
title = Multivariate Analysis
publisher = Academic Press
date = 1979
]

given

:$A sim W_p\left(I, m\right) qquad B sim W_p\left(I, n\right)$

independent and with $m ge p$

:$lambda = frac sim Lambda\left(p,m,n\right).$

The distribution can be related to a product of independent Beta distributed random variables:$u_i sim Bleft\left(frac\left\{m+i-p\right\}\left\{2\right\},frac\left\{p\right\}\left\{2\right\} ight\right)$:$prod_\left\{i=1\right\}^n u_i sim Lambda\left(p,m,n\right).$

In the context of likelihood-ratio tests "m" is typically the error degrees of freedom, and "n" is the hypothesis degrees of freedom, so that $n+m$ is the total degrees of freedom.

For large "m" Bartlett's approximation [cite journal
last = Bartlett
first = M.S.
title = A note on multiplying factors for various $chi^2$ approximations
journal = J. Royal Statist. Soc. Series B
volume = 16
pages = 296–298
date = 1954
] allows Wilks' lambda to be approximated with a Chi-square distribution:$left\left(frac\left\{p-n+1\right\}\left\{2\right\}-m ight\right)log Lambda\left(p,m,n\right) sim chi^2_\left\{np\right\}.$

References

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