Owen's T function explained

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

T(h,a) = (1)/(2 \pi) \int ^{a} _{0} (e ^{- (1)/(h ^{2} ( 1+ x ^{2)} 2)})/(1+ x ^{2}) dx \left( -inf ty < h, a < +inf ty\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 < 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

T(h, 0 ) = 0 T( 0 ,a) = (1)/(2 \pi) \arctan(a) T( - h,a) = T(h,a) T(h, - a) = - T(h,a) T(h,a) + T\left(ah, (1)/(a) \right) = \begin{cases} (1)/(2) \left(\Phi(h) + \Phi(ah)\right) - \Phi(h)\Phi(ah) & if a \geq 0 \ (1)/(2) \left(\Phi(h) + \Phi(ah)\right) - \Phi(h)\Phi(ah) - (1)/(2) & if a < 0 \end{cases} T(h, 1 ) = (1)/(2) \Phi(h) \left( 1 - \Phi(h)\right) \int T( 0 ,x) d x = x T( 0 ,x) - (1)/(4 \pi) ln \left( 1+ x ^{2\right)} + C Here Φ(x) is the standard normal cumulative distribution function\Phi(x) = (1)/(\sqrt{ 2 \pi) } \int_^x \exp\left(-\frac\right) \, \mathrmt More properties can be found in the literature.

References

• Owen . D. . 1980 . A table of normal integrals . Communications in Statistics: Simulation and Computation . 389–419 . B9 . 4 . 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)

Notes and 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. http://people.sc.fsu.edu/~jburkardt/m_src/asa076/tfn.m JC Young and Christoph Minder. Algorithm AS 76