In mathematics, Owen's T function ''T''(''h'', ''a''), named after

statistician A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, and statisticians may wor ...

Donald Bruce Owen, is defined by :

T(h,a)=\frac\int_^ \frac  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

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standard normal In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is ...

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s. This function can be used to calculate

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probabilities and, from there, in the calculation of

multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...

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

T(0,a) = \frac \arctan(a)

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

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

T(h,a) + T\left(ah,\frac\right) = \begin \frac \left(\Phi(h) + \Phi(ah)\right) - \Phi(h)\Phi(ah) &\text \quad a \geq 0 \\  \frac \left(\Phi(h) + \Phi(ah)\right) - \Phi(h)\Phi(ah) - \frac & \text \quad a < 0 \end

T(h, 1) = \frac \Phi(h) \left(1 - \Phi(h)\right)

\int T(0,x) \, \mathrmx = x T(0,x) - \frac \ln\left(1+x^2\right) + C

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

\Phi(x) = \frac \int_^x \exp\left(-\frac\right) \, \mathrmt

More properties can be found in the literature.

References

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 Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimi ...

since version 8, a
OwenT

External links

Why You Should Care about the Obscure
(Wolfram blog post) Normal distribution Computational statistics Functions related to probability distributions {{statistics-stub