A Note on Confidence Intervals for a Binomial p: Andersson–Nerman vs. Wilson

ABSTRACT

Numerous approximate confidence intervals in closed form have been suggested over the years for the parameter in the binomial distribution. One of the oldest and still most advocated is the Wilson (score) interval, which in this article will be compared with the less well‐known Andersson–Nerman (henceforth AN) interval. These intervals are quite closely related, but it will be shown analytically and illustrated by examples that the coverage probability of the AN interval always equals or exceeds that of the Wilson interval, while also having uniformly larger expected length. Asymptotic expressions for the coverage probability and expected length of the AN interval are provided, using Edgeworth and Taylor expansions, respectively. The well‐behaved Wilson and AN pivots are furthermore contrasted with the problematic Wald pivot. The latter gives rise to the Wald interval, which is probably one of the best known and most used procedures altogether in the history of statistical inference. Unfortunately, this interval performs poorly, but even to this day it is to be found in many current textbooks. Therefore, it is still of relevance to search for attractive alternatives based on sound statistical principles.