Owen's T function

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

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

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^[1]^[2] and, from there, in the calculation of multivariate normal distribution probabilities.^[3]

Computer algorithms for the accurate calculation of this function are available.^[4] The function was first introduced by Owen in 1956.^[5]

^ 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.
^ Owen, D B (1956). "Tables for computing bivariate normal probabilities". Ann. Math. Statist., 27, 1075–1090.

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.