Wilks' lambda distribution

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


: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}.


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