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Consider a linear regression model with n-dimensional response vector,p-dimensional regression parameter beta and independent normally distributederrors. Suppose that the parameter of interest is theta = a^T beta where a is aspecified vector. Define the s-dimensional parameter vector tau=C^T beta-twhere C and t are specified. Also suppose that we have uncertain priorinformation that tau=0. Part of our evaluation of a frequentist confidenceinterval for theta is the ratio (expected length of this confidenceinterval)/(expected length of standard 1-alpha confidence interval), the scaledexpected length of this interval. We say that a 1-alpha confidence interval fortheta utilizes this uncertain prior information if (a) the scaled expectedlength of this interval is significantly less than 1 when tau=0, (b) themaximum value of the scaled expected length is not too large and (c) thisconfidence interval reverts to the standard 1-alpha confidence interval whenthe data happen to strongly contradict the prior information. Lethat{Theta}=a^T hat{beta} and hat{tau}=C^T hat{beta}-t, where hat{beta} is theleast squares estimator of beta. We consider the particular case that thatE((hat{tau}-tau)(hat{Theta}-theta))=0, so that hat{Theta} and hat{tau} areindependent. We present a new 1-alpha confidence interval for theta thatutilizes the uncertain prior informationthat tau=0. The following problem isused to illustrate the application of this new confidence interval. Consider a2^3 factorial experiment with 1 replicate. Suppose that the parameter ofinterest theta is a specified linear combination of the main effects. Assumethat the three-factorinteraction is zero. Also suppose that we have uncertainprior information that all of the two-factor interactions are zero. Our aim isto find a frequentist 0.95 confidence interval for theta that utilizes thisuncertain prior information.

On confidence intervals in regression that utilize uncertain prior information about a vector parameter