Frequentist and Bayesian Interval Estimators for the Normal Mean

It is well known that a Bayesian credible interval for a parameter of interest is derived from a prior distribution that appropriately describes the prior information. However, it is less well known that there exists a frequentist approach developed by Pratt (1961 Pratt , J. W. ( 1961 ). Length of confidence intervals . J. Amer. Statist. Assoc. 56 : 549 – 657 .Taylor & Francis Online], Web of Science ®] , Google Scholar]) that also utilizes prior information in the construction of frequentist confidence intervals. This frequentist approach produces confidence intervals that have minimum weighted average expected length, averaged according to some weight function that appropriately describes the prior information. We begin with a simple model as a starting point in comparing these two distinct procedures in interval estimation. Consider X ₁,…, X _n that are independent and identically N(μ, σ²) distributed random variables, where σ² is known, and the parameter of interest is μ. Suppose also that previous experience with similar data sets and/or specific background and expert opinion suggest that μ = 0. Our aim is to: (a) develop two types of Bayesian 1 ? α credible intervals for μ, derived from an appropriate prior cumulative distribution function F(μ) more importantly; (b) compare these Bayesian 1 ? α credible intervals for μ to the frequentist 1 ? α confidence interval for μ derived from Pratt's frequentist approach, in which the weight function corresponds to the prior cumulative distribution function F(μ). We show that the endpoints of the Bayesian 1 ? α credible intervals for μ are very different to the endpoints of the frequentist 1 ? α confidence interval for μ, when the prior information strongly suggests that μ = 0 and the data supports the uncertain prior information about μ. In addition, we assess the performance of these intervals by analyzing their coverage probability properties and expected lengths.