Owen's T function

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

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

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

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, 10751090.
  2. Sowden, R R and Ashford, J R (1969). "Computation of the bivariate normal integral". Applied Statististics, 18, 169180.
  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, 8194.
  5. Patefield, M. and Tandy, D. (2000) "Fast and accurate Calculation of Owen’s T-Function", Journal of Statistical Software, 5 (5), 125.
  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 (4): 389–419. doi: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.


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