# How to derive Clopper-Pearson interval's F and beta approximation?

It is well-known that there is an approximation of the Clopper-Pearson exact Confident Interval for binomial test.

On Wikipedia it just simply claimed, without any reference that:

Because of a relationship between the cumulative binomial distribution and the beta distribution, the Clopper-Pearson interval is sometimes presented in an alternate format that uses quantiles from the beta distribution. $$B\left(\frac{\alpha}{2}; x, n - x + 1\right) < \theta < B\left(1 - \frac{\alpha}{2}; x + 1, n - x\right)$$

And later I found in C-P that this canbe regarded as an interpolation of the binomial c.d.f. due to the CI-belt discrete arguement. But I still have no clue about how it is derived.

$$\left( 1 + \frac{n - x + 1}{x\,\,F\!\left[1 - \frac{1}{2}\alpha; 2x, 2(n - x + 1)\right]} \right)^{-1}< \theta < \left( 1 + \frac{n - x}{\left[x + 1\right]\,F\!\left[\frac{\alpha}{2}; 2(x + 1), 2(n - x)\right]} \right)^{-1}$$

And then Agresti also touched it in his Categorical Data Analysis, 3ed and leave it:

...from connections between binomial sums and the incomplete beta function and related cdf's of beta and F distributions, the confidence interval is...

Now I want to ask for a reference which gives full details about the proof of this approximation form to Clopper-Pearson CI since I have already spent quite a while on it.

FYI:Agresti and C-P did not solve the problem in their papers, I want a paper or a book which fully gives the arguement about the incomplete Beta function calculation since I myself is not familiar with this sort of manipulation.

Thanks.

• math.stackexchange.com/questions/1428803/… math.se copy. – Henry.L Sep 10 '15 at 0:18
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