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.Owen, D B (1956). "Tables for computing bivariate normal probabilities". Annals of Mathematical Statistics,

27, 1075–1090.

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 probabilitiesSowden, 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. and, from there, in the calculation of multivariate normal distribution probabilities.Schervish, M H (1984). "Multivariate normal probabilities with error bound". Applied Statistics, 33, 81–94.

It also frequently appears in various integrals involving Gaussian functions.

Computer algorithms for the accurate calculation of this function are available;Patefield, M. and Tandy, D. (2000) "[http://www.jstatsoft.org/v05/i05/paper Fast and accurate Calculation of Owen’s T-Function]", Journal of Statistical Software, 5 (5), 1–25. quadrature having been employed since the 1970s. [http://people.sc.fsu.edu/~jburkardt/m_src/asa076/tfn.m JC Young and Christoph Minder. Algorithm AS 76]

Properties

: $T(h,0) = 0$

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

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

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

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

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

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

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

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

More properties can be found in the literature.{{cite journal

| last1 = Owen | first1 = D.

| year = 1980

| title = A table of normal integrals

| journal = Communications in Statistics: Simulation and Computation

| pages = 389–419

| volume = B9

| issue = 4

| doi = 10.1080/03610918008812164

}}

References

Software

[https://people.sc.fsu.edu/~jburkardt/m_src/owen/owen.html 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 [http://reference.wolfram.com/mathematica/ref/OwenT.html OwenT].
Owen's T-function is also available as a function in R (https://search.r-project.org/CRAN/refmans/sn/html/T.Owen.html).

External links

[http://blog.wolfram.com/2010/10/07/why-you-should-care-about-the-obscure/ Why You Should Care about the Obscure] (Wolfram blog post)

Category:Normal distribution

Category:Computational statistics

Category:Functions related to probability distributions