Owen's T function

<h2 id="applications">Applications</h2>
The function T(h, a) gives the probability of the event (X > h and 0 < Y < aX) where X and Y are <a href="/facts/Statistically_independent/NUzQtnUL">independent</a> <a href="/facts/Standard_normal_distribution/UapjjPyQ">standard normal</a> <a href="/facts/Random_variable/TwTBXnLT">random variables</a>.
This function can be used to calculate <a href="/facts/Bivariate_normal_distribution/2Xfegqz2">bivariate normal distribution</a> probabilities<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> and, from there, in the calculation of <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">multivariate normal distribution</a> probabilities.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
It also frequently appears in <a href="/facts/List_of_integrals_of_Gaussian_functions/oaGR8NaF">various integrals involving Gaussian functions</a>.
Computer algorithms for the accurate calculation of this function are available;<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> <a href="/facts/Gaussian_quadrature/MLmU1Dgc">quadrature</a> having been employed since the 1970s. <a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="properties">Properties</h2>

T
        (
        h
        ,
        0
        )
        =
        0
      
    
    {\displaystyle T(h,0)=0}

T
        (
        0
        ,
        a
        )
        =
        
          
            1
            
              2
              π
            
          
        
        arctan
        ⁡
        (
        a
        )
      
    
    {\displaystyle T(0,a)={\frac {1}{2\pi }}\arctan(a)}

T
        (
        −
        h
        ,
        a
        )
        =
        T
        (
        h
        ,
        a
        )
      
    
    {\displaystyle T(-h,a)=T(h,a)}

T
        (
        h
        ,
        −
        a
        )
        =
        −
        T
        (
        h
        ,
        a
        )
      
    
    {\displaystyle T(h,-a)=-T(h,a)}

T
 (
 h
 ,
 a
 )
 +
 T
 
 (
 
 a
 h
 ,
 
 
 1
 a
 
 
 
 )
 
 =
 
 
 {
 
 
 
 
 
 1
 2
 
 
 
 (
 
 Φ
 (
 h
 )
 +
 Φ
 (
 a
 h
 )
 
 )
 
 −
 Φ
 (
 h
 )
 Φ
 (
 a
 h
 )
 
 
 
 if
 
 
 a
 ≥
 0
 
 
 
 
 
 
 1
 2
 
 
 
 (
 
 Φ
 (
 h
 )
 +
 Φ
 (
 a
 h
 )
 
 )
 
 −
 Φ
 (
 h
 )
 Φ
 (
 a
 h
 )
 −
 
 
 1
 2
 
 
 
 
 
 if
 
 
 a
 <
 0
 
 
 
 
 
 
 
 
 {\displaystyle T(h,a)+T\left(ah,{\frac {1}{a}}\right)={\begin{cases}{\frac {1}{2}}\left(\Phi (h)+\Phi (ah)\right)-\Phi (h)\Phi (ah)&{\text{if}}\quad a\geq 0\\{\frac {1}{2}}\left(\Phi (h)+\Phi (ah)\right)-\Phi (h)\Phi (ah)-{\frac {1}{2}}&{\text{if}}\quad a<0\end{cases}}}

T
        (
        h
        ,
        1
        )
        =
        
          
            1
            2
          
        
        Φ
        (
        h
        )
        
          (
          
            1
            −
            Φ
            (
            h
            )
          
          )
        
      
    
    {\displaystyle T(h,1)={\frac {1}{2}}\Phi (h)\left(1-\Phi (h)\right)}

∫
        T
        (
        0
        ,
        x
        )
        
        
          d
        
        x
        =
        x
        T
        (
        0
        ,
        x
        )
        −
        
          
            1
            
              4
              π
            
          
        
        ln
        ⁡
        
          (
          
            1
            +
            
              x
              
                2
              
            
          
          )
        
        +
        C
      
    
    {\displaystyle \int T(0,x)\,\mathrm {d} x=xT(0,x)-{\frac {1}{4\pi }}\ln \left(1+x^{2}\right)+C}

Here Φ(x) is the <a href="/facts/Normal_distribution/UapjjPyQ">standard normal cumulative distribution function</a>

Φ
        (
        x
        )
        =
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            −
            ∞
          
          
            x
          
        
        exp
        ⁡
        
          (
          
            −
            
              
                
                  t
                  
                    2
                  
                
                2
              
            
          
          )
        
        
        
          d
        
        t
      
    
    {\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}\exp \left(-{\frac {t^{2}}{2}}\right)\,\mathrm {d} t}

More properties can be found in the literature.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="software">Software</h2>
<ul><li><a href="https://people.sc.fsu.edu/~jburkardt/m_src/owen/owen.html">Owen's T function</a> (user web site) - offers C++, FORTRAN77, FORTRAN90, and MATLAB libraries released under the LGPL license <a href="/facts/LGPL/AXwkz9x3">LGPL</a></li>
<li>Owen's T-function is implemented in <a href="/facts/Mathematica/dRwoGFG2">Mathematica</a> since version 8, as <a href="http://reference.wolfram.com/mathematica/ref/OwenT.html">OwenT</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://blog.wolfram.com/2010/10/07/why-you-should-care-about-the-obscure/">Why You Should Care about the Obscure</a> (Wolfram blog post)</li></ul>

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

<ol>
<li id="fn:1">Owen, D B (1956). "Tables for computing bivariate normal probabilities". Annals of Mathematical Statistics,
27, 1075–1090. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Sowden, R R and Ashford, J R (1969). "Computation of the bivariate normal integral". Applied Statististics, 18, 169–180. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Donelly, T G (1973). "Algorithm 462. Bivariate normal distribution". Commun. Ass. Comput.Mach., 16, 638. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Schervish, M H (1984). "Multivariate normal probabilities with error bound". Applied Statistics, 33, 81–94. <a href="/wiki/Error_bound" target="_blank">/wiki/Error_bound</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Patefield, M. and Tandy, D. (2000) "Fast and accurate Calculation of Owen’s T-Function", Journal of Statistical Software, 5 (5), 1–25.
 <a href="http://www.jstatsoft.org/v05/i05/paper" target="_blank">http://www.jstatsoft.org/v05/i05/paper</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">JC Young and Christoph Minder. Algorithm AS 76
 <a href="http://people.sc.fsu.edu/~jburkardt/m_src/asa076/tfn.m" target="_blank">http://people.sc.fsu.edu/~jburkardt/m_src/asa076/tfn.m</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Owen, D. (1980). "A table of normal integrals". Communications in Statistics: Simulation and Computation. B9 (4): 389–419. doi:10.1080/03610918008812164. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Owen's T function open-in-new

Owen's T function