Locally correct confidence intervals for a binomial proportion: A new criteria for an interval estimator

TL;DR

This paper introduces a new criterion for binomial confidence intervals called 'locally correct', proposes a method that satisfies it with shorter average length than existing methods, and compares it favorably to the Clopper-Pearson and mid-p methods.

Contribution

The paper proposes a novel criterion for confidence intervals for binomial proportions and introduces a new method that optimally satisfies this criterion with shorter intervals.

Findings

The new method produces shorter average-length intervals than traditional methods.

It satisfies the 'locally correct' confidence interval criterion.

Compared to Clopper-Pearson, it offers appreciably smaller average intervals.

Abstract

Well-recommended methods of forming `confidence intervals' for a binomial proportion give interval estimates that do not actually meet the definition of a confidence interval, in that their coverages are sometimes lower than the nominal confidence level. The methods are favoured because their intervals have a shorter average length than the Clopper-Pearson (gold-standard) method, whose intervals really are confidence intervals. Comparison of such methods is tricky -- the best method should perhaps be the one that gives the shortest intervals (on average), but when is the coverage of a method so poor that it should not be classed as a means of forming confidence intervals? As the definition of a confidence interval is not being adhered to, another criterion for forming interval estimates for a binomial proportion is needed. In this paper we suggest a new criterion; methods which meet the criterion are said to yield $\textit{locally correct confidence intervals}$. We propose a method that yields such intervals and prove that its intervals have a shorter average length than those of any other method that meets the criterion. Compared with the Clopper-Pearson method, the proposed method gives intervals with an appreciably smaller average length. The mid-$p$ method also satisfies the new criterion and has its own optimality property.