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

TL;DR

This paper develops a new confidence interval for a linear regression parameter that effectively incorporates uncertain prior information, achieving shorter expected length when the prior is correct while maintaining coverage.

Contribution

It introduces a novel 1-alpha confidence interval that leverages uncertain prior info about a parameter, balancing reduced length with robustness.

Findings

The new interval is shorter when prior info is correct.

It reverts to the standard interval when data contradicts prior.

Application to factorial experiments demonstrates practical utility.

Abstract

Consider a linear regression model with n-dimensional response vector, p-dimensional regression parameter beta and independent normally distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the s-dimensional parameter vector 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), 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 large and (c) this confidence interval reverts to the standard 1-alpha confidence interval when the data happen to strongly contradict the prior information. Let hat{Theta}=a^T hat{beta} and hat{tau}=C^T hat{beta}-t, where hat{beta} is the least squares estimator of beta. We consider the particular case that that E((hat{tau}-tau)(hat{Theta}-theta))=0, so that hat{Theta} and hat{tau} are independent. We present a new 1-alpha confidence interval for theta that utilizes the uncertain prior informationthat tau=0. The following problem is used to illustrate the application of this new confidence interval. Consider a 2^3 factorial experiment with 1 replicate. Suppose that the parameter of interest theta is a specified linear combination of the main effects. Assume that the three-factorinteraction is zero. Also suppose that we have uncertain prior information that all of the two-factor interactions are zero. Our aim is to find a frequentist 0.95 confidence interval for theta that utilizes this uncertain prior information.