Confidence Intervals in Regression That Utilize Uncertain Prior Information About a Vector Parameter

Consider a linear regression model with independent normally distributed errors. Suppose that the scalar parameter of interest is a specified linear combination of the components of the regression parameter vector. Also suppose that we have uncertain prior information that a parameter vector, consisting of specified distinct linear combinations of these components, takes a given value. Part of our evaluation of a frequentist confidence interval for the parameter of interest is the scaled expected length, defined to be the expected length of this confidence interval divided by the expected length of the standard confidence interval for this parameter, with the same confidence coefficient. We say that a confidence interval for the parameter of interest utilizes this uncertain prior information if (a) the scaled expected length of this interval is substantially less than one when the prior information is correct, (b) the maximum value of the scaled expected length is not too large and (c) this confidence interval reverts to the standard confidence interval, with the same confidence coefficient, when the data happen to strongly contradict the prior information. We present a new confidence interval for a scalar parameter of interest, with specified confidence coefficient, that utilizes this uncertain prior information. A factorial experiment with one replicate is used to illustrate the application of this new confidence interval.