A Comparison of Bayesian and Frequentist Interval Estimators in Regression that Utilize Uncertain Prior Information

Abstract

SummaryConsider a linear regression model with independent normally distributed errors. Suppose that the scalar parameter of interest is a specified linear combination of the components of the regression parameter vector. Also suppose that we have uncertain prior information that a distinct specified linear combination of these components takes the value zero. We provide succinct and informative descriptions of interval estimators for the parameter of interest using the new concepts of scaled offset and scaled half‐length. We describe the Bayesian equi‐tailed and shortest credible intervals for the parameter of interest that result from a prior density for the parameter about which we have uncertain prior information that is a mixture of a rectangular ‘slab’ and a Dirac delta function ‘spike’, combined with noninformative prior densities for the other parameters of the model. This prior belongs to the class of ‘slab and spike’ priors, which have been used for Bayesian variable selection. We compare these credible intervals with Kabaila and Giri's frequentist confidence interval for the parameter of interest that utilizes this uncertain prior information. We show that these frequentist and Bayesian interval estimators depend on the data in very different ways. We also consider some close variants of this prior distribution that lead to Bayesian and frequentist interval estimators with greater similarity. Nonetheless, as we show, substantial differences between these interval estimators remain.