# 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,[1]

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.[1]

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[2] allows Wilks's lambda to be approximated with a chi-squared distribution

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

## References

1. ^ a b c Mardia, K.V.; J.T. Kent, J.M. Bibby (1979). Multivariate Analysis. Academic Press.
2. ^ Bartlett, M.S. (1954). "A note on multiplying factors for various $\chi^2$ approximations". Journal of the Royal Statistical Society, Series B 16: 296–298.