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]
independent and with
The distribution can be related to a product of independent Beta distributed random variables::
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 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 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:
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