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 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.

Wilks' lambda distribution is related to two independent Wishart distributed variables, and is defined as follows,

given

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

independent and with $m\geq p$

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

where p is the number of dimensions. 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.

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

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

$\prod _{i=1}^{n}u_{i}\sim \Lambda (p,m,n).$

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

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

See also

• Chi-squared distribution
• F-distribution
• Gamma distribution
• Hotelling's T-squared distribution
• Student's t-distribution
• Wishart distribution