Noncentral beta distribution

<h2 id="cumulative-distribution-function">Cumulative distribution function</h2>
<p>The Type I <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is usually represented as a <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson</a> mixture of central <a href="/facts/Beta_distribution/DbJk4eeV">beta</a> random variables:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>

F
        (
        x
        )
        =
        
          ∑
          
            j
            =
            0
          
          
            ∞
          
        
        P
        (
        j
        )
        
          I
          
            x
          
        
        (
        α
        +
        j
        ,
        β
        )
        ,
      
    
    {\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha +j,\beta ),}

<p>where λ is the noncentrality parameter, <i>P</i>(.) is the Poisson(λ/2) probability mass function, <i>\alpha=m/2</i> and <i>\beta=n/2</i> are shape parameters, and 
  
    
      
        
          I
          
            x
          
        
        (
        a
        ,
        b
        )
      
    
    {\displaystyle I_{x}(a,b)}
  
 is the <a href="/facts/Regularized_incomplete_beta_function/Dcm1JAxb">incomplete beta function</a>. That is,
</p>

F
        (
        x
        )
        =
        
          ∑
          
            j
            =
            0
          
          
            ∞
          
        
        
          
            1
            
              j
              !
            
          
        
        
          
            (
            
              
                λ
                2
              
            
            )
          
          
            j
          
        
        
          e
          
            −
            λ
            
              /
            
            2
          
        
        
          I
          
            x
          
        
        (
        α
        +
        j
        ,
        β
        )
        .
      
    
    {\displaystyle F(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}I_{x}(\alpha +j,\beta ).}

<p>The Type II <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> in mixture form is
</p>

F
        (
        x
        )
        =
        
          ∑
          
            j
            =
            0
          
          
            ∞
          
        
        P
        (
        j
        )
        
          I
          
            x
          
        
        (
        α
        ,
        β
        +
        j
        )
        .
      
    
    {\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha ,\beta +j).}

<p>Algorithms for evaluating the noncentral beta distribution functions are given by Posten<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> and Chattamvelli.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="probability-density-function">Probability density function</h2>
<p>The (Type I) <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> for the noncentral beta distribution is:
</p>

f
        (
        x
        )
        =
        
          ∑
          
            j
            =
            0
          
          
            ∞
          
        
        
          
            1
            
              j
              !
            
          
        
        
          
            (
            
              
                λ
                2
              
            
            )
          
          
            j
          
        
        
          e
          
            −
            λ
            
              /
            
            2
          
        
        
          
            
              
                x
                
                  α
                  +
                  j
                  −
                  1
                
              
              (
              1
              −
              x
              
                )
                
                  β
                  −
                  1
                
              
            
            
              B
              (
              α
              +
              j
              ,
              β
              )
            
          
        
        .
      
    
    {\displaystyle f(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}{\frac {x^{\alpha +j-1}(1-x)^{\beta -1}}{B(\alpha +j,\beta )}}.}

<p>where 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is the <a href="/facts/Beta_function/Dcm1JAxb">beta function</a>, 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 and 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
 are the shape parameters, and 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 is the <a href="/facts/Noncentrality_parameter/uWgoRYTm">noncentrality parameter</a>. The density of <i>Y</i> is the same as that of <i>1-X</i> with the degrees of freedom reversed.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="related-distributions">Related distributions</h2>
<h3>Transformations</h3>
<p>If 
  
    
      
        X
        ∼
        
          
            Beta
          
        
        
          (
          
            α
            ,
            β
            ,
            λ
          
          )
        
      
    
    {\displaystyle X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)}
  
, then 
  
    
      
        
          
            
              β
              X
            
            
              α
              (
              1
              −
              X
              )
            
          
        
      
    
    {\displaystyle {\frac {\beta X}{\alpha (1-X)}}}
  
 follows a <a href="/facts/Noncentral_F-distribution/YCyc6Bus">noncentral F-distribution</a> with 
  
    
      
        2
        α
        ,
        2
        β
      
    
    {\displaystyle 2\alpha ,2\beta }
  
 degrees of freedom, and non-centrality parameter 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
.
</p><p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 follows a <a href="/facts/Noncentral_F-distribution/YCyc6Bus">noncentral F-distribution</a> 
  
    
      
        
          F
          
            
              μ
              
                1
              
            
            ,
            
              μ
              
                2
              
            
          
        
        
          (
          λ
          )
        
      
    
    {\displaystyle F_{\mu _{1},\mu _{2}}\left(\lambda \right)}
  
 with 
  
    
      
        
          μ
          
            1
          
        
      
    
    {\displaystyle \mu _{1}}
  
 numerator degrees of freedom and 
  
    
      
        
          μ
          
            2
          
        
      
    
    {\displaystyle \mu _{2}}
  
 denominator degrees of freedom, then
</p>

Z
        =
        
          
            
              
                
              
              
                
                  
                    
                      
                        
                      
                      
                        
                          
                            μ
                            
                              2
                            
                          
                        
                      
                    
                    
                      
                        
                      
                      
                        
                          
                            μ
                            
                              1
                            
                          
                        
                      
                    
                  
                
              
            
            
              
                
              
              
                
                  
                    
                      
                        
                          
                        
                        
                          
                            
                              μ
                              
                                2
                              
                            
                          
                        
                      
                      
                        
                          
                        
                        
                          
                            
                              μ
                              
                                1
                              
                            
                          
                        
                      
                    
                  
                  +
                  
                    X
                    
                      −
                      1
                    
                  
                
              
            
          
        
      
    
    {\displaystyle Z={\cfrac {\cfrac {\mu _{2}}{\mu _{1}}}{{\cfrac {\mu _{2}}{\mu _{1}}}+X^{-1}}}}

<p>follows a noncentral Beta distribution:
</p>

Z
        ∼
        
          
            Beta
          
        
        
          (
          
            
              
                1
                2
              
            
            
              μ
              
                1
              
            
            ,
            
              
                1
                2
              
            
            
              μ
              
                2
              
            
            ,
            λ
          
          )
        
      
    
    {\displaystyle Z\sim {\mbox{Beta}}\left({\frac {1}{2}}\mu _{1},{\frac {1}{2}}\mu _{2},\lambda \right)}
  
.
<p>This is derived from making a straightforward transformation.
</p>
<h3>Special cases</h3>
<p>When 
  
    
      
        λ
        =
        0
      
    
    {\displaystyle \lambda =0}
  
, the noncentral beta distribution is equivalent to the (central) <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a>.
</p>

<h3>Citations</h3>

<h3>Sources</h3>

<ul><li><a href="/facts/Milton_Abramowitz/6yRyez7P">M. Abramowitz</a> and <a href="/facts/Irene_Stegun/FjSS1LDR">I. Stegun</a>, editors (1965) "<a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed">Handbook of Mathematical Functions</a>", Dover: New York, NY.</li>
<li>Hodges, J.L. Jr (1955). <a href="https://doi.org/10.1214%2Faoms%2F1177728424">"On the noncentral beta-distribution"</a>. <i>Annals of Mathematical Statistics</i>. 26 (4): 648–653. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faoms%2F1177728424">10.1214/aoms/1177728424</a>.</li>
<li>Seber, G.A.F. (1963). "The non-central chi-squared and beta distributions". <i><a href="/facts/Biometrika/hg97ERcX">Biometrika</a></i>. 50 (3–4): 542–544. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fbiomet%2F50.3-4.542">10.1093/biomet/50.3-4.542</a>.</li>
<li>Christian Walck, "Hand-book on Statistical Distributions for experimentalists."</li></ul>

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

<ol>
<li id="fn:1"><p>Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Posten, H.O. (1993). "An Effective Algorithm for the Noncentral Beta Distribution Function". The American Statistician. 47 (2): 129–131. doi:10.1080/00031305.1993.10475957. JSTOR 2685195. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Noncentral beta distribution open-in-new

Noncentral beta distribution