Construction of fixed width confidence intervals for a Bernoulli success probability using sequential sampling: a simulation study

This article considers the construction of level 1 - alpha fixed width 2d confidence intervals for a Bernoulli success probability p, assuming no prior knowledge about p and so p can be anywhere in the interval [0, 1]. It is shown that some fixed width 2d confidence intervals that combine sequential sampling of Hall [Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Stat. 9 (1981), pp. 1229-1238] and fixed-sample-size confidence intervals of Agresti and Coull [Approximate is better than 'exact' for interval estimation of binomial proportions, Am. Stat. 52 (1998), pp. 119-126], Wilson [Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927), pp. 209-212] and Brown et al. [Interval estimation for binomial proportion (with discussion), Stat. Sci. 16 (2001), pp. 101-133] have close to 1 - alpha confidence level. These sequential confidence intervals require a much smaller sample size than a fixed-sample-size confidence interval. For the coin jamming example considered, a fixed-sample-size confidence interval requires a sample size of 9457, while a sequential confidence interval requires a sample size that rarely exceeds 2042.