Wilks' lambda distribution

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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. 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 is defined as follows,^[1] given

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

independent and with $m \ge p$

$\lambda = \frac{|A|}{|A+B|} = \frac{1}{|I+A^{-1}B|} \sim \Lambda(p,m,n)$ .

The distribution can be related to a product of independent Beta distributed random variables

$u_i \sim B\left(\frac{m+i-p}{2},\frac{p}{2}\right)$

$\prod_{i=1}^n u_i \sim \Lambda(p,m,n)$

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 degress of freedom.^[1]

For large m Bartlett's approximation ^[2] allows Wilks' lambda to be approximated with a Chi-square distribution

$\left(\frac{p-n+1}{2}-m\right)\log \Lambda(p,m,n) \sim \chi^2_{np}$ .^[1]

[edit] References

^ ^a ^b ^c Mardia, K.V.; J.T. Kent, J.M. Bibby (1979). Multivariate Analysis. Academic Press.
^ Bartlett, M.S. (1954). "A note on multiplying factors for various $χ 2$ approximations". J. Royal Statist. Soc. Series B 16: 296-298.

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Wilks' lambda distribution

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