Noncentral F-distribution

<h2 id="occurrence-and-specification">Occurrence and specification</h2>
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Noncentral_chi-squared_distribution/saAcjyRh">noncentral chi-squared</a> random variable with noncentrality parameter 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 and 
 
 
 
 
 ν
 
 1
 
 
 
 
 {\displaystyle \nu _{1}}
 
 degrees of freedom, and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared</a> random variable with 
 
 
 
 
 ν
 
 2
 
 
 
 
 {\displaystyle \nu _{2}}
 
 degrees of freedom that is <a href="/facts/Statistical_independence/NUzQtnUL">statistically independent</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
, then

F
        =
        
          
            
              X
              
                /
              
              
                ν
                
                  1
                
              
            
            
              Y
              
                /
              
              
                ν
                
                  2
                
              
            
          
        
      
    
    {\displaystyle F={\frac {X/\nu _{1}}{Y/\nu _{2}}}}

is a noncentral F-distributed random variable.
The <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (pdf) for the noncentral F-distribution is<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

p
        (
        f
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          
            
              
                e
                
                  −
                  λ
                  
                    /
                  
                  2
                
              
              (
              λ
              
                /
              
              2
              
                )
                
                  k
                
              
            
            
              B
              
                (
                
                  
                    
                      
                        ν
                        
                          2
                        
                      
                      2
                    
                  
                  ,
                  
                    
                      
                        ν
                        
                          1
                        
                      
                      2
                    
                  
                  +
                  k
                
                )
              
              k
              !
            
          
        
        
          
            (
            
              
                
                  ν
                  
                    1
                  
                
                
                  ν
                  
                    2
                  
                
              
            
            )
          
          
            
              
                
                  ν
                  
                    1
                  
                
                2
              
            
            +
            k
          
        
        
          
            (
            
              
                
                  ν
                  
                    2
                  
                
                
                  
                    ν
                    
                      2
                    
                  
                  +
                  
                    ν
                    
                      1
                    
                  
                  f
                
              
            
            )
          
          
            
              
                
                  
                    ν
                    
                      1
                    
                  
                  +
                  
                    ν
                    
                      2
                    
                  
                
                2
              
            
            +
            k
          
        
        
          f
          
            
              ν
              
                1
              
            
            
              /
            
            2
            −
            1
            +
            k
          
        
      
    
    {\displaystyle p(f)=\sum \limits _{k=0}^{\infty }{\frac {e^{-\lambda /2}(\lambda /2)^{k}}{B\left({\frac {\nu _{2}}{2}},{\frac {\nu _{1}}{2}}+k\right)k!}}\left({\frac {\nu _{1}}{\nu _{2}}}\right)^{{\frac {\nu _{1}}{2}}+k}\left({\frac {\nu _{2}}{\nu _{2}+\nu _{1}f}}\right)^{{\frac {\nu _{1}+\nu _{2}}{2}}+k}f^{\nu _{1}/2-1+k}}

when 
 
 
 
 f
 ≥
 0
 
 
 {\displaystyle f\geq 0}
 
 and zero otherwise.
The degrees of freedom 
 
 
 
 
 ν
 
 1
 
 
 
 
 {\displaystyle \nu _{1}}
 
 and 
 
 
 
 
 ν
 
 2
 
 
 
 
 {\displaystyle \nu _{2}}
 
 are positive.
The term 
 
 
 
 B
 (
 x
 ,
 y
 )
 
 
 {\displaystyle B(x,y)}
 
 is the <a href="/facts/Beta_function/Dcm1JAxb">beta function</a>, where

B
        (
        x
        ,
        y
        )
        =
        
          
            
              Γ
              (
              x
              )
              Γ
              (
              y
              )
            
            
              Γ
              (
              x
              +
              y
              )
            
          
        
        .
      
    
    {\displaystyle B(x,y)={\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}.}

The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> for the noncentral F-distribution is

F
        (
        x
        ∣
        
          d
          
            1
          
        
        ,
        
          d
          
            2
          
        
        ,
        λ
        )
        =
        
          ∑
          
            j
            =
            0
          
          
            ∞
          
        
        
          (
          
            
              
                
                  
                    (
                    
                      
                        
                          1
                          2
                        
                      
                      λ
                    
                    )
                  
                  
                    j
                  
                
                
                  j
                  !
                
              
            
            
              e
              
                −
                λ
                
                  /
                
                2
              
            
          
          )
        
        I
        
          (
          
            
              
                
                  
                    d
                    
                      1
                    
                  
                  x
                
                
                  
                    d
                    
                      2
                    
                  
                  +
                  
                    d
                    
                      1
                    
                  
                  x
                
              
            
            
              
                |
              
            
            
              
                
                  d
                  
                    1
                  
                
                2
              
            
            +
            j
            ,
            
              
                
                  d
                  
                    2
                  
                
                2
              
            
          
          )
        
      
    
    {\displaystyle F(x\mid d_{1},d_{2},\lambda )=\sum \limits _{j=0}^{\infty }\left({\frac {\left({\frac {1}{2}}\lambda \right)^{j}}{j!}}e^{-\lambda /2}\right)I\left({\frac {d_{1}x}{d_{2}+d_{1}x}}{\bigg |}{\frac {d_{1}}{2}}+j,{\frac {d_{2}}{2}}\right)}

where 
 
 
 
 I
 
 
 {\displaystyle I}
 
 is the <a href="/facts/Regularized_incomplete_beta_function/Dcm1JAxb">regularized incomplete beta function</a>.
The mean and variance of the noncentral F-distribution are

E
        ⁡
        [
        F
        ]
        
        
          
            {
            
              
                
                  =
                  
                    
                      
                        
                          ν
                          
                            2
                          
                        
                        (
                        
                          ν
                          
                            1
                          
                        
                        +
                        λ
                        )
                      
                      
                        
                          ν
                          
                            1
                          
                        
                        (
                        
                          ν
                          
                            2
                          
                        
                        −
                        2
                        )
                      
                    
                  
                
                
                  
                    if 
                  
                  
                    ν
                    
                      2
                    
                  
                  >
                  2
                
              
              
                
                  
                    does not exist
                  
                
                
                  
                    if 
                  
                  
                    ν
                    
                      2
                    
                  
                  ≤
                  2
                
              
            
            
          
        
      
    
    {\displaystyle \operatorname {E} [F]\quad {\begin{cases}={\frac {\nu _{2}(\nu _{1}+\lambda )}{\nu _{1}(\nu _{2}-2)}}&{\text{if }}\nu _{2}>2\\{\text{does not exist}}&{\text{if }}\nu _{2}\leq 2\\\end{cases}}}

and

Var
        ⁡
        [
        F
        ]
        
        
          
            {
            
              
                
                  =
                  2
                  
                    
                      
                        (
                        
                          ν
                          
                            1
                          
                        
                        +
                        λ
                        
                          )
                          
                            2
                          
                        
                        +
                        (
                        
                          ν
                          
                            1
                          
                        
                        +
                        2
                        λ
                        )
                        (
                        
                          ν
                          
                            2
                          
                        
                        −
                        2
                        )
                      
                      
                        (
                        
                          ν
                          
                            2
                          
                        
                        −
                        2
                        
                          )
                          
                            2
                          
                        
                        (
                        
                          ν
                          
                            2
                          
                        
                        −
                        4
                        )
                      
                    
                  
                  
                    
                      (
                      
                        
                          
                            ν
                            
                              2
                            
                          
                          
                            ν
                            
                              1
                            
                          
                        
                      
                      )
                    
                    
                      2
                    
                  
                
                
                  
                    if 
                  
                  
                    ν
                    
                      2
                    
                  
                  >
                  4
                
              
              
                
                  
                    does not exist
                  
                
                
                  
                    if 
                  
                  
                    ν
                    
                      2
                    
                  
                  ≤
                  4.
                
              
            
            
          
        
      
    
    {\displaystyle \operatorname {Var} [F]\quad {\begin{cases}=2{\frac {(\nu _{1}+\lambda )^{2}+(\nu _{1}+2\lambda )(\nu _{2}-2)}{(\nu _{2}-2)^{2}(\nu _{2}-4)}}\left({\frac {\nu _{2}}{\nu _{1}}}\right)^{2}&{\text{if }}\nu _{2}>4\\{\text{does not exist}}&{\text{if }}\nu _{2}\leq 4.\\\end{cases}}}

<h2 id="special-cases">Special cases</h2>
When λ = 0, the noncentral F-distribution becomes the
<a href="/facts/F-distribution/XNuuJtWY">F-distribution</a>.

<h2 id="related-distributions">Related distributions</h2>
Z has a <a href="/facts/Noncentral_chi-squared_distribution/saAcjyRh">noncentral chi-squared distribution</a> if

Z
        =
        
          lim
          
            
              ν
              
                2
              
            
            →
            ∞
          
        
        
          ν
          
            1
          
        
        F
      
    
    {\displaystyle Z=\lim _{\nu _{2}\to \infty }\nu _{1}F}

where F has a noncentral F-distribution.
See also <a href="/facts/Noncentral_t-distribution/KumJDJFe">noncentral t-distribution</a>.

<h2 id="implementations">Implementations</h2>
The noncentral F-distribution is implemented in the <a href="/facts/R_(programming_language)/LSrkr8K8">R</a> language (e.g., pf function), in <a href="/facts/MATLAB/qPjLISCk">MATLAB</a> (ncfcdf, ncfinv, ncfpdf, ncfrnd and ncfstat functions in the statistics toolbox) in <a href="/facts/Mathematica/dRwoGFG2">Mathematica</a> (NoncentralFRatioDistribution function), in <a href="/facts/NumPy/T6FnhWWD">NumPy</a> (random.noncentral_f), and in <a href="/facts/Boost_C%252B%252B_Libraries/vGLGG51A">Boost C++ Libraries</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
A collaborative wiki page implements an interactive online calculator, programmed in the <a href="/facts/R_(programming_language)/LSrkr8K8">R language</a>, for the noncentral t, <a href="/facts/Noncentral_chi-squared_distribution/saAcjyRh">chi-squared</a>, and F distributions, at the Institute of Statistics and Econometrics of the <a href="/facts/Humboldt_University_of_Berlin/QE2lhhkd">Humboldt University of Berlin</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a>; et al. <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html">"Noncentral F-distribution"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>. Wolfram Research, Inc. Retrieved 20 August 2011.</li></ul>

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

<ol>
<li id="fn:1">Kay, S. (1998). Fundamentals of Statistical Signal Processing: Detection Theory. New Jersey: Prentice Hall. p. 29. ISBN 0-13-504135-X. <a href="0-13-504135-X" target="_blank">0-13-504135-X</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">John Maddock; Paul A. Bristow; Hubert Holin; Xiaogang Zhang; Bruno Lalande; Johan Råde. "Noncentral F Distribution: Boost 1.39.0". Boost.org. Retrieved 20 August 2011. <a href="http://www.boost.org/doc/libs/1_39_0/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html" target="_blank">http://www.boost.org/doc/libs/1_39_0/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Sigbert Klinke (10 December 2008). "Comparison of noncentral and central distributions". Humboldt-Universität zu Berlin. <a href="http://mars.wiwi.hu-berlin.de/mediawiki/slides/index.php/Comparison_of_noncentral_and_central_distributions" target="_blank">http://mars.wiwi.hu-berlin.de/mediawiki/slides/index.php/Comparison_of_noncentral_and_central_distributions</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Noncentral F-distribution open-in-new

Noncentral F-distribution