Wilks's 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 is defined as follows,

given

$A\sim W_{p}(I,m)\qquad B\sim W_{p}(I,n)$

independent and with $m\geq p$

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

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 degrees of freedom.

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

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