Paul Kabaila, Khageswor Giri
Abstract
Consider a linear regression model with n-dimensional response vector, regression parameter β= (β₁, ..., β_p) and independent and identically N(0, σ²) distributed errors. Suppose that the parameter of interest is θ= a^T βwhere a is a specified vector. Define the parameter τ= c^T 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 θis the ratio (expected length of this confidence interval)/(expected length of standard 1-αconfidence interval), which we call the scaled expected length of this interval. We say that a 1-αconfidence interval for θutilizes 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-αconfidence 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 βdenote the least squares estimator of β. Also let Θ= a^T βand τ= c^T β- t. Using computations and new theoretical results, we show that the performance of this confidence interval improves as |Corr(Θ, τ)| increases and n-p decreases.