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 ) = 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.[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

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.[7]

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 (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.

External links

  • Why You Should Care about the Obscure (Wolfram blog post)


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