Confidence intervals for the normal mean utilizing prior information

We consider $X_1,X_2,...,X_n$ that are independent and identically $N(\mu,\sigma^2)$ distributed random variables. Suppose that the parameter of interest is $\mu$ and also suppose that we have uncertain prior information that $\mu/\sigma$ is close to 0. Our aim is to find a frequentist $1-\alpha$ confidence interval for $\mu$ that utilizes this prior information. Pratt and Brown, Casella and Hwang have described such confidence intervals for the cases that $\sigma^2$ is known and that $\sigma^2$ is unknown respectively. These confidence intervals have the major problem that if the prior information happens to be badly incorrect (i.e. $\mu/\sigma$ happens to be far away from 0) then these confidence intervals have very large expected lengths. In this paper we find $1-\alpha$ confidence intervals for $\mu$ that do not suffer from this problem. For the case that $\sigma^2$ is known we do this by extending the method used by Pratt. For the case that $\sigma^2$ is unknown we do this using a new methodology. Our $1-\alpha$ confidence intervals have the following desirable properties. They have expected lengths that (a) are relatively small when the prior information about $\mu/\sigma$ is correct and (b) have a maximum value that is not too large. They also coincide with the corresponding standard confidence interval when the data happens to strongly contradict the prior information about $\mu/\sigma$.