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 isdefined as follows,cite book
last = Mardia
first = K.V.
coauthors = J.T. Kent, J.M. Bibby
title = Multivariate Analysis
publisher = Academic Press
date = 1979]

given

:$A\; sim\; W\_p(I,\; m)\; qquad\; B\; sim\; W\_p(I,\; n)$

independent and with $m\; ge\; p$

:$lambda\; =\; frac\; sim\; Lambda(p,m,n).$

The distribution can be related to a product of independent Beta distributed random variables:$u\_i\; sim\; Bleft(frac\{m+i-p\}\{2\},frac\{p\}\{2\}\; ight)$:$prod\_\{i=1\}^n\; u\_i\; sim\; Lambda(p,m,n).$

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 [cite journal
last = Bartlett
first = M.S.
title = A note on multiplying factors for various $chi^2$ approximations
journal = J. Royal Statist. Soc. Series B
volume = 16
pages = 296–298
date = 1954] allows Wilks' lambda to be approximated with a Chi-square distribution:$left(frac\{p-n+1\}\{2\}-m\; ight)log\; Lambda(p,m,n)\; sim\; chi^2\_\{np\}.$

References