Interval estimation of a population mean using existing knowledge or data on effect sizes

Bayes or empirical Bayes methods to improve inferential accuracy for a population mean has been widely adopted in medical research. As the joint prior distribution of both the mean and variance parameters can be difficult to specify or estimate, most of these methods have relied on certain level of simplifications of the joint prior, which could lead to difficulty in the interpretation of the posterior distribution or compromised inferential accuracy. We propose a framework of interval estimation using existing knowledge or data on the effect size to address this difficulty. Our method has two unique characteristics. First, the interpretation of the interval bears the spirit of both Frequentist and Bayesian thinking. For this reason, it will be called FB interval. Second, we define a new quantity, the hybrid effect size, which is a key quantity that mediates the construction of the FB interval when the population variance is unknown. A simulation study and a real data example are presented to evaluate and illustrate our method.