Wilks's lambda distribution

In statistics, Wilks's 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 (MANOVA). It is a multivariate generalization of the univariate F-distribution, generalizing the F-distribution in the same way that the Hotelling's T-squared distribution generalizes Student's t-distribution.

Wilks's 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 $m \ge p$

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

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

$u_i \sim B\left(\frac{m+1-i}{2},\frac{n}{2}\right)$

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

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

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

See also

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