Instatistics,Wilks' lambda distribution (named forSamuel S. Wilks), is aprobability distribution used inmultivariatehypothesis testing, especially with regard to thelikelihood-ratio test andmultivariate analysis of variance (MANOVA).

Wilks' lambda distribution is defined from twoindependentWishart distributed variables as theratio distribution of theirdeterminants,^[1]

given

${\displaystyle \mathbf{A}\sim W_p(\mathrm{\Sigma },m)\mathbf{B}\sim W_p(\mathrm{\Sigma },n)}$

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

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

wherep is the number of dimensions. In the context oflikelihood-ratio testsm is typically the error degrees of freedom, andn is the hypothesis degrees of freedom, so that${\displaystyle n+m}$ is the total degrees of freedom.^[1]

There is a symmetry among the parameters of the Wilks distribution,^[1]

${\displaystyle \mathrm{\Lambda }(p,m,n)\sim \mathrm{\Lambda }(n,m+n-p,p)}$

Computations or tables of the Wilks' distribution for higher dimensions are not readily available and one usually resorts to approximations.One approximation is attributed toM. S. Bartlett and works for largem^[2] allows Wilks' lambda to be approximated with achi-squared distribution

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

Another approximation is attributed toC. R. Rao.^[1][3]

The distribution can be related to a product ofindependentbeta-distributed random variables

${\displaystyle u_i\sim B\left(\frac{m+i-p}{2},\frac{p}{2}\right)}$

${\displaystyle \prod _{i=1}^n{u}_i\sim \mathrm{\Lambda }(p,m,n).}$

As such it can be regarded as a multivariate generalization of the beta distribution.

It follows directly that for a one-dimension problem, when the Wishart distributions are one-dimensional with${\displaystyle p=1}$ (i.e., chi-squared-distributed), then the Wilks' distribution equals the beta-distribution with a certain parameter set,

${\displaystyle \mathrm{\Lambda }(1,m,n)\sim B\left(\frac{m}{2},\frac{n}{2}\right).}$

From the relations between a beta and anF-distribution, Wilks' lambda can be related to the F-distribution when one of the parameters of the Wilks lambda distribution is either 1 or 2, e.g.,^[1]

${\displaystyle \frac{1-\mathrm{\Lambda }(p,m,1)}{\mathrm{\Lambda }(p,m,1)}\sim \frac{p}{m-p+1}{F}_{p,m-p+1},}$

and

${\displaystyle \frac{1-\sqrt{\mathrm{\Lambda }(p,m,2)}}{\sqrt{\mathrm{\Lambda }(p,m,2)}}\sim \frac{p}{m-p+1}{F}_{2p,2(m-p+1)}.}$

• Continuous distributions

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