Owen's T function
In mathematics, Owen's T function T(h, a), named after statistician Donald Bruce Owen, is defined by
- [math]\displaystyle{ 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 \lt h, a \lt +\infty\right). }[/math]
The function was first introduced by Owen in 1956.[1]
Applications
The function T(h, a) gives the probability of the event (X > h and 0 < Y < aX) where X and Y are independent standard normal random variables.
This function can be used to calculate bivariate normal distribution probabilities[2][3] and, from there, in the calculation of multivariate normal distribution probabilities.[4] It also frequently appears in various integrals involving Gaussian functions.
Computer algorithms for the accurate calculation of this function are available;[5] quadrature having been employed since the 1970s. [6]
Properties
- [math]\displaystyle{ T(h,0) = 0 }[/math]
- [math]\displaystyle{ T(0,a) = \frac{1}{2\pi} \arctan(a) }[/math]
- [math]\displaystyle{ T(-h,a) = T(h,a) }[/math]
- [math]\displaystyle{ T(h,-a) = -T(h,a) }[/math]
- [math]\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 \lt 0 \end{cases} }[/math]
- [math]\displaystyle{ T(h, 1) = \frac{1}{2} \Phi(h) \left(1 - \Phi(h)\right) }[/math]
- [math]\displaystyle{ \int T(0,x) \, \mathrm{d}x = x T(0,x) - \frac{1}{4 \pi} \ln\left(1+x^2\right) + C }[/math]
Here Φ(x) is the standard normal cumulative distribution function
- [math]\displaystyle{ \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp\left(-\frac{t^2 }{2}\right) \, \mathrm{d}t }[/math]
More properties can be found in the literature.[7]
References
- ↑ Owen, D B (1956). "Tables for computing bivariate normal probabilities". Annals of Mathematical Statistics, 27, 1075–1090.
- ↑ Sowden, R R and Ashford, J R (1969). "Computation of the bivariate normal integral". Applied Statististics, 18, 169–180.
- ↑ Donelly, T G (1973). "Algorithm 462. Bivariate normal distribution". Commun. Ass. Comput.Mach., 16, 638.
- ↑ Schervish, M H (1984). "Multivariate normal probabilities with error bound". Applied Statistics, 33, 81–94.
- ↑ Patefield, M. and Tandy, D. (2000) "Fast and accurate Calculation of Owen’s T-Function", Journal of Statistical Software, 5 (5), 1–25.
- ↑ JC Young and Christoph Minder. Algorithm AS 76
- ↑ (Owen 1980)
- Owen, D. (1980). "A table of normal integrals". Communications in Statistics: Simulation and Computation B9 (4): 389–419. doi:10.1080/03610918008812164.
Software
- Owen's T function (user web site) - offers C++, FORTRAN77, FORTRAN90, and MATLAB libraries released under the LGPL license LGPL
- Owen's T-function is implemented in Mathematica since version 8, as OwenT.
External links
- Why You Should Care about the Obscure (Wolfram blog post)
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