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

摘要

Consider a linear regression model with regression parameter beta andnormally distributed errors. Suppose that the parameter of interest is theta =a^T beta where a is a specified vector. Define the parameter tau = c^T beta - twhere c and t are specified and a and c are linearly independent. Also supposethat we have uncertain prior information that tau = 0. Kabaila and Giri, 2009,JSPI, describe a new frequentist 1-alpha confidence interval for theta thatutilizes this uncertain prior information. We compare this confidence intervalwith Bayesian 1-alpha equi-tailed and shortest credible intervals for thetathat result from a prior density for tau that is a mixture of a rectangular"slab" and a Dirac delta function "spike", combined with noninformative priordensities for the other parameters of the model. We show that these frequentistand Bayesian interval estimators depend on the data in very different ways. Wealso consider some close variants of this prior distribution that lead toBayesian and frequentist interval estimators with greater similarity.Nonetheless, as we show, substantial differences between these intervalestimators remain.