Further Properties of Frequentist Confidence Intervals in Regression that Utilize Uncertain Prior Information

Consider a linear regression model with n‐dimensional response vector, regression parameter β=(β1,…,βp) and independent and identically N(0,σ2) distributed errors. Suppose that the parameter of interest is θ=a⊤β where a is a specified vector. Define the parameter τ=c⊤β−t where c and t are specified. Also suppose that we have uncertain prior information that τ=0. Part of our evaluation of a frequentist confidence interval for θ is the ratio (expected length of this confidence interval)/(expected length of standard 1−α confidence interval), which we call the scaled expected length of this interval. We say that a 1−α confidence interval for θ utilizes this uncertain prior information if: (i) the scaled expected length of this interval is substantially less than 1 when τ=0; (ii) the maximum value of the scaled expected length is not too much larger than 1; and (iii) this confidence interval reverts to the standard 1−α confidence interval when the data happen to strongly contradict the prior information. Kabaila and Giri (2009) present a new method for finding such a confidence interval. Let β̂ denote the least squares estimator of β. Also let Θ̂=a⊤β̂ and τ̂=c⊤β̂−t. Using computations and new theoretical results, we show that the performance of this confidence interval improves as |corr(Θ̂,τ̂)| increases and n−p decreases.