Owen's Function: A Simple Solution to Complex Problems

Definition

 

If you are interested in statistics or probability, you may have encountered the Owen's function, a special function that arises in various applications involving bivariate normal distributions. Let’s explain what Owen's function is, how it is defined and notated, and why it is useful. The Owen's function is defined as follows:

for any real numbers h and a, Owen's function T(h,a) is given by
$T(h,a)=\frac{1}{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}·\; <!--\; middle\; dot\; -->\underset{0}{\overset{a}{\int \; <!--\; integral\; -->}}\frac{\exp (-\frac{h^2}{2}·\; <!--\; middle\; dot\; -->(1+x^2))}{1+x^2}\mathrm{dx}$
where h and a are any real numbers. The Owen's function can be interpreted as the probability of the event (X > h and 0 < Y < a * X), where X and Y are independent standard normal random variables. This means that it can be used to calculate the area under the bivariate normal density curve in a certain region. In other words, it is the probability that (X,Y) lies in a wedge-shaped region bounded by the lines x = h, y = 0 and y = ax.

Owen's function has many applications in statistics and probability. For instance, it appears in the calculation of the cumulative distribution function of the noncentral t-distribution, the power of certain statistical tests, the multivariate normal tail probabilities, and the multivariate normal orthant probabilities. It also has connections to copulas, elliptical distributions, and directional statistics.

Background

Owen's function is a special function that arises in the context of multivariate normal integrals. It was first introduced by Donald Bruce Owen in 1956, who was interested in the problem of testing the equality of two normal means when the variances are unknown and unequal. Owen derived an expression for the power function of this test, which involved a double integral that could not be evaluated in closed form. He then proposed a numerical approximation for this integral, based on a series expansion of a function that he called T(h,a), where h and a are real parameters.

Owen's function has many interesting properties and applications, such as:

 - It is symmetric in h and a, i.e., T(h,a) = T(a,h). 

- It is related to the standard normal cumulative distribution function $\mathrm{\Phi \; <!--greek\; capital\; letter\; phi-->}\left(z\right)$ by $T(h,0)=\frac{1}{2}\mathrm{\Phi \; <!--greek\; capital\; letter\; phi-->}(-h)$ and $T(0,a)=\frac{1}{2\mathrm{\pi \; <!--greek\; small\; letter\; pi-->}}·\; <!--middle\; dot-->\arctan \left(a\right)$
- It satisfies the recurrence relation $T(h,a)=T(h,0)-T(h,\sqrt{a^2+h^2})$
- It can be used to compute the probability of a rectangular region under a bivariate normal distribution.
- It can be used to compute the tail probabilities of certain quadratic forms in normal variables.

Applications

One of the most useful applications of Owen's function is in computing bivariate and multivariate normal probabilities. In this section, I will explain how Owen's function can be used to calculate the probability that two or more normally distributed variables fall within a given region. Let X and Y be two independent standard normal variables, and let A and B be two constants. The bivariate normal probability P(X < A, Y < B) can be expressed as a function of Owen's function T(h,a), where h = B/A and a = A. This result was first derived by Owen (1956) and can be written as:

$P(X<A,Y<B)=\mathrm{\Phi \; <!--greek\; capital\; letter\; phi-->}\left(A\right)·\; <!--middle\; dot-->\mathrm{\Phi \; <!--greek\; capital\; letter\; phi-->}\left(B\right)-T(h,a)-T(a,h)$

where Φ is the standard normal cumulative distribution function. This formula allows us to compute bivariate normal probabilities without using numerical integration or tables.

The formula can be generalized to multivariate normal probabilities as well. Let X1, X2, ..., Xn be n independent standard normal variables, and let A1, A2, ..., An be n constants. The multivariate normal probability P(X1 < A1, X2 < A2, ..., Xn < An) can be expressed as a sum of products of Owen's functions $T(h_i,a_i)$, where $h_i=A_i/A_j$ and $a_i=A_j$ for i ≠ j. This result was first derived by Genz (1992) and can be written as:

$P(X_1<A_1,\dots \; <!--horizontal\; ellipsis-->,X_n<A_n)=\underset{k=1}{\overset{n}{\sum \; <!--n-ary\; summation-->}}-1^k·\; <!--middle\; dot-->\mathrm{\Phi \; <!--greek\; capital\; letter\; phi-->}\left(A_k\right)·\; <!--middle\; dot-->\underset{i\ne \; <!--not\; equal\; to-->k}{\prod \; <!--n-ary\; product-->}T(h_i,a_i)$

This formula allows us to compute multivariate normal probabilities without using numerical integration or tables. Owen's function is therefore a powerful tool for calculating bivariate and multivariate normal probabilities in various fields of statistics and applied mathematics. Some examples of its applications include testing hypotheses, evaluating integrals, estimating parameters, and simulating random vectors.

Software

The Owen’s function is used to compute the bivariate normal distribution function and related quantities, including the distribution function of a skew-normal variate. The Owen’s function is evaluated using the OwenQ package in R. The package provides two functions for evaluating the Owen’s function: OwenT(h, a) and OwenQ(h, a). The OwenT(h, a) function evaluates the Owen T-function while the OwenQ(h, a) function evaluates the Owen Q-function. You can install the package using the following command:

install.packages("OwenQ")

Here is an example of how to use the OwenT(h, a) function:

library(OwenQ)
OwenT(1.5, 2)

This will evaluate the Owen T-function with h=1.5 and a=2. In addition to the OwenQ package in R that I mentioned earlier, there are several other packages available for computing the Owen’s function in R. These include the CompQuadForm package and the mvtnorm package. In Matlab, you can use the skewt function in the Statistics and Machine Learning Toolbox to compute the cdf of a skew normal distribution. However, Matlab does not have an implementation of Owen’s T-function in its statistical toolbox.

References

Here is a list of recommended reading to learn more about the topic:

1.           Owen, D. B. (1956). Tables for computing bivariate normal probabilities. The Annals of Mathematical Statistics, 27(4), 1075-1090.

2.           Owen, D. B. (1980). A table of normal integrals. Communications in Statistics-Simulation and Computation, 9(4), 389-419.

3.           Owen, D. B., & Rabinowitz, M. (1983). A handbook of the Owen function and related functions. CRC Press.

4.           Owen, D. B., & Zhou, Y. (1990). Safe computation of probability integrals of the multivariate normal and multivariate t distributions. Statistics & Probability Letters, 9(4), 307-311.

5.           Genz A., (1992). Numerical computation of multivariate normal probabilities. Journal of computational and graphical statistics, 1(2):141-149.

6.           Genz, A., & Bretz, F. (2002). Methods for the computation of multivariate t-probabilities. Journal of Computational and Graphical Statistics, 11(4), 950-971.

7.           Genz, A., & Bretz, F. (2009). Computation of multivariate normal and t probabilities. Springer Science & Business Media.

8.           Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., Scheipl, F., & Hothorn, T. (2020). mvtnorm: Multivariate Normal and t Distributions. R package version 1.1-1.

9.           Phadia, E. G. (2010). A survey of the theory of hypergeometric functions of several variables. In Hypergeometric functions on domains of positivity, Jack polynomials, and applications (pp. 25-53). American Mathematical Soc.

10.       Phadia, E. G., & Srivastava, H. M. (2012). Some generalizations and applications of the Owen function T (h; a). Integral Transforms and Special Functions, 23(8), 575-588.

11.       Srivastava, H. M., & Daoust, M.-C. (1991). Some families of the multivariable H-functions with applications to probability distributions. Journal of Statistical Planning and Inference, 29(1), 11-26.

12.       Srivastava, H. M., & Gupta, K. C. (1982). The H-functions of one and two variables with applications. South Asian Publishers.

13.       Srivastava, H. M., & Karlsson, P. W. (1985). Multiple Gaussian hypergeometric series (Vol. 49). Ellis Horwood Limited.

14.       Srivastava, H.M., Choi J., Agarwal P., Jain S.K.(Eds.) (2018) Advances in Special Functions and Orthogonal Polynomials: Proceedings of the International Conference on Special Functions: Theory Computation and Applications held at Indian Institute of Technology Delhi during December 19–23 2016 Springer Singapore

15.       NIST Digital Library of Mathematical Functions (DLMF) https://dlmf.nist.gov/

16.       Wolfram MathWorld http://mathworld.wolfram.com/

17.       Wolfram Functions Site http://functions.wolfram.com/

18.       Abramowitz M., Stegun I.A.(Eds.) (1964) Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables Dover Publications

19.       Gradshteyn I.S., Ryzhik I.M.(Eds.) (2007) Table of Integrals Series and Products Elsevier

20.       Prudnikov A.P., Brychkov Y.A., Marichev O.I.(Eds.) (1998) Integrals and Series CRC Press

21.       Erdélyi A.(Ed.) (1953) Higher Transcendental Functions McGraw-Hill

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