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 is defined as follows,

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 degrees of freedom.

For large m Bartlett's approximation allows Wilks' lambda to be approximated with a Chi-square distribution
 * $$\left(\frac{p-n+1}{2}-m\right)\log \Lambda(p,m,n) \sim \chi^2_{np}.$$