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.


  • $\begingroup$ math.stackexchange.com/questions/1428803/… math.se copy. $\endgroup$ – Henry.L Sep 10 '15 at 0:18
  • $\begingroup$ The on topic help asks that we don't cross post; it requests that we choose one best location and post there. If you later decide a different site is better, you can flag to have it migrated. $\endgroup$ – Glen_b Sep 10 '15 at 3:45

The using of beta distribution to construct the limits of the Biomial Proportions (Clopper-Pearson Formula) is ilustrated in the article "Confidence Bounds & Intervals for Parameters Relating to the Binomial, Negative Binomial, Poisson and Hypergeometric Distributions" by Fritz Scholz (2008). You can get it from:

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  • $\begingroup$ While this link may theoretically answer the question, the help on how to answer questions under "Provide context for links" asks that we "quote the most relevant part of an important link"; as such it would be preferable to include the essential parts of the answer here, and provide the link for reference. $\endgroup$ – Glen_b May 25 '17 at 0:50

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