Further properties of frequentist confidence intervals in regression that utilize uncertain prior information
Abstract
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 = aT β where a is a specified vector. Define the parameter τ = cT 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 (a) the scaled expected length of this interval is significantly less than 1 when τ = 0, (b) the maximum value of the scaled expected length is not too much larger than 1 and (c) 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, JSPI present a new method for finding such a confidence interval. Let β denote the least squares estimator of β. Also let = aT β and τ = cT β - t. Using computations and new theoretical results, we show that the performance of this confidence interval improves as |Corr(, τ)| increases and n-p decreases.