Binomial inference is one of the foundational problems of statistical inference. What can be said about the probability of an event given a fixed number of trials, n, in which the event has "succeeded'' (i.e., occurred) k times? Clopper-Pearson bounds are the canonical frequentist approach to binomial inference, but the two-sided Clopper-Pearson bounds are notoriously overconservative. That is, they are wider than necessary.
For over fifty years, there has been a small but persistent effort to improve on the Clopper-Pearson bounds without relaxing the coverage probability requirement. However, these attempts have all resulted in confidence regions with odd pathologies. Some of them result in spiky p-value distributions, leading to confidence regions that aren't simple intervals. Others result in confidence intervals that aren't nested for different confidence levels.
Dr. Balch recently (2020) overcame these difficulties, producing a unimodal consonant confidence structure for binomial inference. This structure produces confidence regions that form a nested sequence, are always simple intervals, and almost always tighter than the corresponding Clopper-Pearson bounds. This successfully completes a line of inquiry first opened by Theodore Sterne in 1954.Some R-scripts are attached below. The script entitled binomialCCS.R includes the basic functions necessary to implement this new solution. The function "binomial_pvalue" produces the two-sided p-value (i.e., plausibility) of any hypothetical value of the underlying probability being inferred. For those familiar with possibility theory or confidence structures, this function can be applied to a range of probability values, producing a pointwise possiblity (i.e., plausibility) function, which is the most compact way of representing a consonant confidence structure or (more generally) a possibility distribution. For those more inclined to traditional statistics, the function "binomial_ConfInt" yields two-sided confidence intervals at the user-specified level. It can even be applied to yield confidence intervals for multiple simultaneous confidence levels. Finally, the function "critThetaWalley" is an internal function that supports the other two. It is only meant for users who take a serious interest in the theoretical underpinnings of this solution. The other R-scripts below are part of the supporting documentation for a forthcoming paper on this work. |

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