Wilks's lambda distribution

<h2 id="definitions">Definitions</h2>
Wilks' lambda distribution is defined from two <a href="/facts/Statistical_independence/NUzQtnUL">independent</a> <a href="/facts/Wishart_distribution/MEcqTump">Wishart distributed</a> variables as the <a href="/facts/Ratio_distribution/5rUEmWkG">ratio distribution</a> of their <a href="/facts/Determinant/eGYqJ2TF">determinants</a>,<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
given

A
        
        ∼
        
          W
          
            p
          
        
        (
        Σ
        ,
        m
        )
        
        
          B
        
        ∼
        
          W
          
            p
          
        
        (
        Σ
        ,
        n
        )
      
    
    {\displaystyle \mathbf {A} \sim W_{p}(\Sigma ,m)\qquad \mathbf {B} \sim W_{p}(\Sigma ,n)}

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

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

where p is the number of dimensions. In the context of <a href="/facts/Likelihood-ratio_test/oFqLh3R0">likelihood-ratio tests</a> m is typically the error degrees of freedom, and n is the hypothesis degrees of freedom, so that 
 
 
 
 n
 +
 m
 
 
 {\displaystyle n+m}
 
 is the total degrees of freedom.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h2 id="properties">Properties</h2>
There is a symmetry among the parameters of the Wilks distribution,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

Λ
        (
        p
        ,
        m
        ,
        n
        )
        ∼
        Λ
        (
        n
        ,
        m
        +
        n
        −
        p
        ,
        p
        )
      
    
    {\displaystyle \Lambda (p,m,n)\sim \Lambda (n,m+n-p,p)}

<h3>Approximations</h3>
Computations or tables of the Wilks' distribution for higher dimensions are not readily available and one usually resorts to approximations.
One approximation is attributed to <a href="/facts/M._S._Bartlett/RkaCTTwR">M. S. Bartlett</a> and works for large m<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> allows Wilks' lambda to be approximated with a <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a>

(
 
 
 
 
 p
 −
 n
 +
 1
 
 2
 
 
 −
 m
 
 )
 
 log
 ⁡
 Λ
 (
 p
 ,
 m
 ,
 n
 )
 ∼
 
 χ
 
 n
 p
 
 
 2
 
 
 .
 
 
 {\displaystyle \left({\frac {p-n+1}{2}}-m\right)\log \Lambda (p,m,n)\sim \chi _{np}^{2}.}
 
<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Another approximation is attributed to <a href="/facts/C._R._Rao/SxnbA5YV">C. R. Rao</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="related-distributions">Related distributions</h2>
The distribution can be related to a product of <a href="/facts/Statistical_independence/NUzQtnUL">independent</a> <a href="/facts/Beta_distribution/DbJk4eeV">beta-distributed</a> random variables

u
          
            i
          
        
        ∼
        B
        
          (
          
            
              
                
                  m
                  +
                  i
                  −
                  p
                
                2
              
            
            ,
            
              
                p
                2
              
            
          
          )
        
      
    
    {\displaystyle u_{i}\sim B\left({\frac {m+i-p}{2}},{\frac {p}{2}}\right)}

∏
          
            i
            =
            1
          
          
            n
          
        
        
          u
          
            i
          
        
        ∼
        Λ
        (
        p
        ,
        m
        ,
        n
        )
        .
      
    
    {\displaystyle \prod _{i=1}^{n}u_{i}\sim \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 
 
 
 
 p
 =
 1
 
 
 {\displaystyle p=1}
 
 (i.e., chi-squared-distributed), then the Wilks' distribution equals the beta-distribution with a certain parameter set,

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

From the relations between a beta and an <a href="/facts/F-distribution/XNuuJtWY">F-distribution</a>, 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.,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

1
              −
              Λ
              (
              p
              ,
              m
              ,
              1
              )
            
            
              Λ
              (
              p
              ,
              m
              ,
              1
              )
            
          
        
        ∼
        
          
            p
            
              m
              −
              p
              +
              1
            
          
        
        
          F
          
            p
            ,
            m
            −
            p
            +
            1
          
        
        ,
      
    
    {\displaystyle {\frac {1-\Lambda (p,m,1)}{\Lambda (p,m,1)}}\sim {\frac {p}{m-p+1}}F_{p,m-p+1},}

and

1
              −
              
                
                  Λ
                  (
                  p
                  ,
                  m
                  ,
                  2
                  )
                
              
            
            
              Λ
              (
              p
              ,
              m
              ,
              2
              )
            
          
        
        ∼
        
          
            p
            
              m
              −
              p
              +
              1
            
          
        
        
          F
          
            2
            p
            ,
            2
            (
            m
            −
            p
            +
            1
            )
          
        
        .
      
    
    {\displaystyle {\frac {1-{\sqrt {\Lambda (p,m,2)}}}{\sqrt {\Lambda (p,m,2)}}}\sim {\frac {p}{m-p+1}}F_{2p,2(m-p+1)}.}

<ul><li><a href="/facts/Chi-squared_distribution/wYnWWgUe">Chi-squared distribution</a></li>
<li><a href="/facts/Dirichlet_distribution/Qq13f5g2">Dirichlet distribution</a></li>
<li><a href="/facts/F-distribution/XNuuJtWY">F-distribution</a></li>
<li><a href="/facts/Gamma_distribution/lczcdJmw">Gamma distribution</a></li>
<li><a href="/facts/Hotelling%2527s_T-squared_distribution/RJCmedIp">Hotelling's T-squared distribution</a></li>
<li><a href="/facts/Student%2527s_t-distribution/DeT1SDqH">Student's t-distribution</a></li>
<li><a href="/facts/Wishart_distribution/MEcqTump">Wishart distribution</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kanti Mardia, John T. Kent and John Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9. <a href="0-12-471250-9" target="_blank">0-12-471250-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Kanti Mardia, John T. Kent and John Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9. <a href="0-12-471250-9" target="_blank">0-12-471250-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kanti Mardia, John T. Kent and John Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9. <a href="0-12-471250-9" target="_blank">0-12-471250-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">M. S. Bartlett (1954). "A Note on the Multiplying Factors for Various 
 
 
 
 
 χ
 
 2
 
 
 
 
 {\displaystyle \chi ^{2}}
 
 Approximations". J R Stat Soc Ser B. 16 (2): 296–298. JSTOR 2984057. <a href="/wiki/M._S._Bartlett" target="_blank">/wiki/M._S._Bartlett</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Kanti Mardia, John T. Kent and John Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9. <a href="0-12-471250-9" target="_blank">0-12-471250-9</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Kanti Mardia, John T. Kent and John Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9. <a href="0-12-471250-9" target="_blank">0-12-471250-9</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">C. R. Rao (1951). "An Asymptotic Expansion of the Distribution of Wilks' Criterion". Bulletin de l'Institut International de Statistique. 33: 177–180. <a href="/wiki/C._R._Rao" target="_blank">/wiki/C._R._Rao</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kanti Mardia, John T. Kent and John Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9. <a href="0-12-471250-9" target="_blank">0-12-471250-9</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Wilks's lambda distribution open-in-new

Wilks's lambda distribution