Owen \(T\)-function

The Owen \(T\)-function is defined by \[ T(h,a) = \frac{1}{2\pi}\int_0^a \frac{e^{-\frac12 h^2(1+x^2)}}{1+x^2}\mathrm{d}x \] for \(a, h \in \mathbb{R}\). It corresponds to a certain probability and then its value always lies between \(0\) and \(1\).

Below we compare some obtained values of \(T(h, a)\) to the ones given by Wolfram up to \(20\) digits:

> import OwenT
> owenT 0.1 0.1   - 0.01578338051718359918
3.469446951953614e-18
> owenT 1 0.5     - 0.04306469112078536563
6.938893903907228e-18
> owenT 0.5 0.1   - 0.01399302023628015424
1.734723475976807e-18
> owenT 0.5 0.9   - 0.10007270175061385845
0.0
> owenT 5 0.5     - 0.00000014192549621069
-2.6903883987164715e-18
> owenT 2 0.99999 - 0.01111626714677311317
-4.683753385137379e-17

The owenT function allows infinite values of \(h\) and \(a\):

> import OwenT
> import Data.Number.Erf 
> -- zero: 
> owenT (1/0) 3
0.0
> owenT (-1/0) 3
0.0
> -- equality:
> owenT 1 (1/0)
7.932762696572854e-2
> (1 - normcdf 1)/2
7.932762696572854e-2

The algorithm runs as follows. If \(0 \leq a \leq 1\), and \(0 \leq h \leq 8\), a series expansion is used, the one given by Owen (1956). The series is truncated after the \(50\)-th term. If \(0\leq a \leq 1\), and \(h \geq 8\), an asymptotic approximation is used. Otherwise, the properties of the Owen \(T\)-function are exploited to come down to the case \(0 \leq a \leq 1\). The main property involves the Gaussian cumulative function.

As shown by Owen (1956), the Owen \(T\)-function can be used to evaluate the cumulative distribution function of the bivariate Gaussian distribution.

Non-central Student distribution

The pStudent function of the owen library evaluates the cumulative distribution function of the non-central Student distribution with an integer number of degrees of freedom. For odd values, the algorithm resorts to the Owen \(T\)-function. This algorithm is the one given by Owen (1965).

Below we compare some values to the ones given by R:

> import Student
> pStudent 0.5 2 2.5 - 0.022741814853305026
6.259229246019515e-14
> pStudent 0.5 3 2.5 - 0.022741355255468675
1.8677073776451891e-13

Owen Q-function

The first Owen \(Q\)-function is defined by \[ Q_1(\nu, t, \delta, R) = \frac{1}{\Gamma\left(\frac{\nu}{2}\right)2^{\frac12(\nu-2)}} \int_0^R \Phi\left(\frac{tx}{\sqrt{\nu}}-\delta\right) x^{\nu-1} e^{-\frac{x^2}{2}} \mathrm{d}x \] for \(\nu >0\), \(t, \delta \in \mathbb{R}\) and \(R>0\).

Its value always lies between \(0\) and \(1\).

It is implemented in the owen library for integer values of \(\nu\), under the name owenQ1. The algorithm used is the one given by Owen (1965).

The results are theoretically exact for even values of \(\nu\), up to the fact that the function resorts to the Gaussian cumulative function. For odd values of \(\nu\), it also resorts to the Owen \(T\)-function.

Below we compare some obtained values of \(Q_1(\nu, t, \delta, R)\) to the ones given by Wolfram up to \(20\) digits:

> import OwenQ
> owenQ1 1 3 2 5  - 0.52485658843054409291
2.220446049250313e-16
> owenQ1 2 3 2 5  - 0.62938193306526904118
-1.1102230246251565e-16
> owenQ1 9 3 2 5  - 0.77030437685878389173
-1.1102230246251565e-16
> owenQ1 10 3 2 5 - 0.77383547873740988537
1.1102230246251565e-16

The owenQ1 function also returns a value for \(t,\delta=\pm \infty\):

> import OwenQ
> import Math.Gamma as G
> -- zero:
> owenQ1 3 (-1/0) 2 2 
0.0
> owenQ1 3 2 (1/0) 2  
0.0
> -- equalities:
> owenQ1 5 (1/0) 9 3
0.8909358420502276
> owenQ1 5 (1/0) 99 3
0.8909358420502276
> owenQ1 5 99 (-1/0) 3
0.8909358420502276
> owenQ1 5 9 (-1/0) 3
0.8909358420502276
> G.p (5/2) (3^2/2)
0.8909358420502276

References

• Owen, D. B. (1956). Tables for computing bivariate normal probabilities. Ann. Math. Statist. 27, 1075-90.

• Owen, D. B. (1965). A Special Case of a Bivariate Non-Central \(t\)-Distribution. Biometrika 52, 437-446.