Higher-order approximations for interval estimation in binomial settings

Abstract

In this paper we revisit the classical problem of interval estimation for one-binomial parameter and for the log odds ratio of two binomial parameters. We examine the confidence intervals provided by two versions of the modified log likelihood root: the usual Barndorff-Nielsen's r * and a Bayesian version of the r * test statistic. For the one-binomial problem, this work updates the findings of Brown et al. [2003. Interval estimation in exponential families. Statistica Sinica 13, 19–49; 2002. Confidence intervals for a binomial proportion and asymptotic expansion. The Annals of Statistics 30, 160–201] and Cai [2005. One-sided confidence intervals in discrete distributions. Journal of Statistical Planning and Inference 131, 63–88] to higher-order methods. For the log odds ratio of two binomial parameters we show via Edgeworth expansion that both versions of the r * statistics give confidence intervals which nearly completely eliminate the systematic bias in the unconditional smooth coverage probability. We also give expansions for the length of the confidence intervals.