Consider a linear regression model with n-dimensional response vector, regression parameter beta = (beta_1, ..., beta_p) and independent and identically N(0, sigma^2) 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 - t where c and t are specified. Also suppose that we have uncertain prior information that tau = 0. Part of our evaluation of a frequentist confidence interval for theta is the ratio (expected length of this confidence interval)/(expected length of standard 1-alpha confidence interval), which we call the scaled expected length of this interval. We say that a 1-alpha confidence interval for theta utilizes this uncertain prior information if (a) the scaled expected length of this interval is significantly less than 1 when tau = 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-alpha 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 hatbeta denote the least squares estimator of beta. Also let hatTheta = a^T hatbeta and hattau = c^T hatbeta - t. Using computations and new theoretical results, we show that the performance of this confidence interval improves as |Corr(hatTheta, hattau)| increases and n-p decreases.
Download