Beta inequality symmetries

I was thinking about the work I did when I worked in biostatistics at MD Anderson. This work was practical rather than mathematically elegant, useful in its time but not of long-term interest. However, one result came out of this work that I would call elegant, and that was a symmetry I found.

Let X be a beta(a, b) random variable and let Y be a beta(c, d) random variable. Let g(a, b, c, d) be the probability that a sample from X is larger than a sample from Y.

g(a, b, c, d) = Prob(X > Y)

This function often appeared in the inner loop of a simulation and so we spent thousands of CPU-hours computing its values. I looked for ways to evaluate this function more quickly, and along the way I found a symmetry.

The function I call g was studied by W. R. Thompson in 1933 [1]. Thompson noted two symmetries:

g(a, b, c, d) = 1 - g(c, d, a, b)

and

g(a, b, c, d) = g(d, c, b, a)

I found an additional symmetry:

g(a, b, c, d) = g(d, b, c, a).

The only reference to this result in a journal article as far as I know is a paper I wrote with Saralees Nadarajah [2]. That paper cites an MD Anderson technical report which is no longer online, but I saved a copy here.

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[1] W. R. Thompson. On the Likelihood that One Unknown Probability Exceeds Another in View of the Evidence of Two Samples. Biometrika, Volume 25, Issue 4. pp. 285 - 294.

[2] John D. Cook and Saralees Nadarajah. Stochastic Inequality Probabilities for Adaptively Randomized Clinical Trials. Biometrical Journal. 48 (2006) pp 256-365.

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