Approximate Confidence Intervals for a Binomial p—Once Again

Abstract

The problem of constructing a reasonably simple yet well-behaved confidence interval for a binomial parameter p is old but still fascinating and surprisingly complex. During the last century, many alternatives to the poorly behaved standard Wald interval have been suggested. It seems though that the Wald interval is still much in use in spite of many efforts over the years through publications to point out its deficiencies. This paper constitutes yet another attempt to provide an alternative and it builds on a special case of a general technique for adjusted intervals primarily based on Wald type statistics. The main idea is to construct an approximate pivot with uncorrelated, or nearly uncorrelated, components. The resulting AN (Andersson–Nerman) interval, as well as a modification thereof, is compared with the well-renowned Wilson and AC (Agresti–Coull) intervals and the subsequent discussion will in itself hopefully shed some new light on this seemingly elementary interval estimation situation. Generally, an alternative to the Wald interval is to be judged not only by performance, its expression should also indicate why we will obtain a better behaved interval. It is argued that the well-behaved AN interval meets this requirement.