Owen's T function

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

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

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 < 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

${\displaystyle T(h,0)=0}$

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

${\displaystyle T(-h,a)=T(h,a)}$

${\displaystyle T(h,-a)=-T(h,a)}$

${\displaystyle T(h,a)+T\left(ah,{\frac {1}{a}}\right)={\begin{cases}{\frac {1}{2}}\left(\Phi (h)+\Phi (ah)\right)-\Phi (h)\Phi (ah)&{\text{if}}\quad a\geq 0\\{\frac {1}{2}}\left(\Phi (h)+\Phi (ah)\right)-\Phi (h)\Phi (ah)-{\frac {1}{2}}&{\text{if}}\quad a<0\end{cases}}}$

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

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

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

${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}\exp \left(-{\frac {t^{2}}{2}}\right)\,\mathrm {d} t}$

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