RMSE-minimizing confidence intervals for the binomial parameter

Abstract

Let X be the number of successes in n mutually independent and identically distributed Bernoulli trials, each with probability of success p. For fixed n and $\alpha$, there are $n + 1$ distinct two-sided $100(1 - \alpha )$% confidence intervals for p associated with the outcomes ${X = 0, 1, 2, \ldots , n}$. There is no known exact non-randomized confidence interval for p. Existing approximate confidence interval procedures use a formula, which often requires numerical methods to implement, to calculate confidence interval bounds. The bounds associated with these confidence intervals correspond to discontinuities in the actual coverage function. The paper does not aim to provide a formula for the confidence interval bounds, but rather to select the confidence interval bounds that minimize the root mean square error of the actual coverage function for sample size n and significance level $\alpha$ in the frequentist context.

Appendix

The structure of the algorithm for constructing the 100$(1 - \alpha )$% RMSE-minimizing confidence interval is given below. The four existing confidence interval procedures used in the smoothing and the large-sample portions of the algorithm are the Wilson–score, Jeffreys, Arcsine, and Agresti–Coull.

This algorithm has been implemented in R in the binomTestMSE function in the conf package, which consists of about 700 lines of code. The numerical solution that minimizes the RMSE is performed by the uniroot.all function in R.