# 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,[1]

given

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

independent and with ${\displaystyle m\geq p}$

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

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

${\displaystyle u_{i}\sim B\left({\frac {m+i-p}{2}},{\frac {p}{2}}\right)}$
${\displaystyle \prod _{i=1}^{n}u_{i}\sim \Lambda (p,m,n).}$

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 ${\displaystyle n+m}$ is the total degrees of freedom.[1]

For large m Bartlett's approximation [2] allows Wilks' lambda to be approximated with a Chi-square distribution

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

## References

1. Mardia, K.V. (1979). Multivariate Analysis. Academic Press. Unknown parameter |coauthors= ignored (help)
2. Bartlett, M.S. (1954). "A note on multiplying factors for various ${\displaystyle \chi ^{2}}$ approximations". J. Royal Statist. Soc. Series B. 16: 296–298. Check date values in: |date= (help)