Owen's T function

In mathematics, Owen's T function T(h, a), named after statistician Donald Bruce Owen, is defined by

$T(h,a)=\frac{1}{2\pi}\int_{0}^{a} \frac{e^{-\frac{1}{2} h^2 (1+x^2)}}{1+x^2} dx \quad \left(-\infty < h, a < +\infty\right).$

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<a*X) 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

$T(h,0) = 0$

$T(0,a) = \frac{1}{2\pi} \arctan(a)$

$T(-h,a) = T(h,a)$

$T(h,-a) = -T(h,a)$

$T(h,a) + T(ah,\frac{1}{a}) = \frac{1}{2} \left(\Phi(h) + \Phi(ah)\right) - \Phi(h)\Phi(ah) \quad \mbox{if} \quad a \geq 0$

$T(h,a) + T(ah,\frac{1}{a}) = \frac{1}{2} \left(\Phi(h) + \Phi(ah)\right) - \Phi(h)\Phi(ah) - \frac{1}{2} \quad \mbox{if} \quad a < 0$

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

$\int T(0,x) \mathrm{d}x = x T(0,x) - \frac{1}{4 \pi} \ln(1+x^2) + C$

Here Φ(x) is the standard normal cumulative density function

$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp\left(-t^2 / 2\right) \mathrm{d}y$

More properties can be found in the literature.^[7]

References

1. Owen, D B (1956). "Tables for computing bivariate normal probabilities". Annals of Mathematical Statistics, 27, 1075–1090.
2. Sowden, R R and Ashford, J R (1969). "Computation of the bivariate normal integral". Applied Statististics, 18, 169–180.
3. Donelly, T G (1973). "Algorithm 462. Bivariate normal distribution". Commun. Ass. Comput.Mach., 16, 638.
4. Schervish, M H (1984). "Multivariate normal probabilities with error bound". Applied Statistics, 33, 81–94.
5. Patefield, M. and Tandy, D. (2000) "Fast and accurate Calculation of Owen’s T-Function", Journal of Statistical Software, 5 (5), 1–25.
6. JC Young and Christoph Minder. Algorithm AS 76
7. Owen (1980)

• Owen, D. (1980). "A table of normal integrals". Communications in Statistics: Simulation and Computation B9: 389–419.

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)