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| In mathematics, '''Owen's T function''' ''T''(''h'', ''a''), named after [[statistician]] Donald Bruce Owen, is defined by
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| :<math>
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| T(h,a)=\frac{1}{2\pi}\int_{0}^{a} \frac{e^{-\frac{1}{2} h^2 (1+x^2)}}{1+x^2} dx \quad \left(-\infty < h, a < +\infty\right).
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| </math>
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| ==Applications==
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| The function ''T''(''h'', ''a'') gives the probability of the event (''X>h'' and 0''<Y<a*X'') where ''X'' and ''Y'' are [[statistically independent|independent]] [[standard normal distribution|standard normal]] [[random variable]]s.
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| This function can be used to calculate [[bivariate normal distribution]] probabilities<ref>Sowden, R R and Ashford, J R (1969). "Computation of the bivariate normal integral". ''Applied Statististics'', 18, 169–180.</ref><ref>Donelly, T G (1973). "Algorithm 462. Bivariate normal distribution". ''Commun. Ass. Comput.Mach.'', 16, 638.</ref> and, from there, in the calculation of [[multivariate normal distribution]] probabilities.<ref>Schervish, M H (1984). "Multivariate normal probabilities with [[error bound]]". ''Applied Statistics'', 33, 81–94.</ref>
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| It also frequently appears in [[List of integrals of Gaussian functions|various integrals involving Gaussian functions]].
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| Computer algorithms for the accurate calculation of this function are available.<ref>Patefield, M. and Tandy, D. (2000) "[http://www.jstatsoft.org/v05/i05/paper Fast and accurate Calculation of Owen’s T-Function]", ''Journal of Statistical Software'', 5 (5), 1–25.
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| </ref> The function was first introduced by Owen in 1956.<ref>Owen, D B (1956). "Tables for computing bivariate normal probabilities". ''Annals of Mathematical Statistics'',
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| 27, 1075–1090.</ref>
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| ==Properties==
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| : <math> T(h,0) = 0 </math>
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| : <math> T(0,a) = \frac{1}{2\pi} \arctan(a) </math>
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| : <math> T(-h,a) = T(h,a) </math>
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| : <math> T(h,-a) = -T(h,a) </math>
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| : <math> T(h,a) + T(ah,\frac{1}{a}) = \frac{1}{2} \left(\Phi(h) + \Phi(ah)\right) + \Phi(h)\Phi(ah) \quad \mbox{if} \quad a \geq 0 </math>
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| : <math> T(h,a) + T(ah,\frac{1}{a}) = \frac{1}{2} \left(\Phi(h) + \Phi(ah)\right) + \Phi(h)\Phi(ah) - \frac{1}{2} \quad \mbox{if} \quad a < 0 </math>
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| : <math> T(h, 1) = \frac{1}{2} \Phi(h) \left(1 - \Phi(h)\right) </math>
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| : <math> \int T(0,x) \mathrm{d}x = x T(0,x) - \frac{1}{4 \pi} \ln(1+x^2) + C </math>
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| More properties can be found in the literature.<ref>{{harvtxt|Owen|1980}}</ref>
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| ==References==
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| {{reflist}}
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| * {{cite journal
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| | last1 = Owen | first1 = D.
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| | year = 1980
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| | title = A table of normal integrals
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| | journal = Communications in Statistics: Simulation and Computation
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| | pages = 389–419
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| | volume = B9
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| | ref = harv
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| }}
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| ==Software==
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| * [http://people.sc.fsu.edu/~burkardt/f_src/owens/owens.html Owen's T function] (user web site) - offers C++, FORTRAN77, FORTRAN90, and MATLAB libraries released under the LGPL license [[LGPL]]
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| * Owen's T-function is implemented in [[Mathematica]] since version 8, as [http://reference.wolfram.com/mathematica/ref/OwenT.html OwenT].
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| ==External links==
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| * [http://blog.wolfram.com/2010/10/07/why-you-should-care-about-the-obscure/ Why You Should Care about the Obscure] (Wolfram blog post)
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| [[Category:Multivariate statistics]]
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| [[Category:Computational statistics]]
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| [[Category:Functions related to probability distributions]]
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| {{statistics-stub}}
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