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.
Best Answer
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:
http://www.stat.washington.edu/fritz/DATAFILES498B2008/ConfidenceBounds.pdf