Confidence intervals in regression utilizing prior information

We consider a linear regression model with regression parameter β=(β1,…,βp)β=(β1,…,βp) and independent and identically N(0,σ2)N(0,σ2) distributed errors. Suppose that the parameter of interest is θ=aTβθ=aTβ where aa is a specified vector. Define the parameter τ=cTβ-tτ=cTβ-t where the vector cc and the number tt are specified and aa and cc are linearly independent. Also suppose that we have uncertain prior information that τ=0τ=0. We present a new frequentist 1-α1-α confidence interval for θθ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α1-α confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0τ=0. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about ττ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×22×2 factorial experiment with 20 replicates. Suppose that the parameter of interest θθ is a specified simple   effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θθ that utilizes this prior information.