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Owen's T function

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

T ( h , a ) = 1 2 π ∫ 0 a e − 1 2 h 2 ( 1 + x 2 ) 1 + x 2 d x ( − ∞ < h , a < + ∞ ) . {\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.

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 and, from there, in the calculation of multivariate normal distribution probabilities. It also frequently appears in various integrals involving Gaussian functions.

Computer algorithms for the accurate calculation of this function are available; quadrature having been employed since the 1970s.

Properties

T ( h , 0 ) = 0 {\displaystyle T(h,0)=0}
T ( 0 , a ) = 1 2 π arctan ⁡ ( a ) {\displaystyle T(0,a)={\frac {1}{2\pi }}\arctan(a)}
T ( − h , a ) = T ( h , a ) {\displaystyle T(-h,a)=T(h,a)}
T ( h , − a ) = − T ( h , a ) {\displaystyle T(h,-a)=-T(h,a)}
T ( h , a ) + T ( a h , 1 a ) = { 1 2 ( Φ ( h ) + Φ ( a h ) ) − Φ ( h ) Φ ( a h ) if a ≥ 0 1 2 ( Φ ( h ) + Φ ( a h ) ) − Φ ( h ) Φ ( a h ) − 1 2 if a < 0 {\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}}}
T ( h , 1 ) = 1 2 Φ ( h ) ( 1 − Φ ( h ) ) {\displaystyle T(h,1)={\frac {1}{2}}\Phi (h)\left(1-\Phi (h)\right)}
∫ T ( 0 , x ) d x = x T ( 0 , x ) − 1 4 π ln ⁡ ( 1 + x 2 ) + C {\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

Φ ( x ) = 1 2 π ∫ − ∞ x exp ⁡ ( − t 2 2 ) d t {\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.

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. JC Young and Christoph Minder. Algorithm AS 76
  7. 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.

External links


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