Finding the MVUE of a two-sided interval of a normal

[rayge]

Our task is to determine if P(-c \le X \le c) has a minimum variance unbiased estimator for a sample from a distribution that is N(\theta,1). The one-sided interval P(X \le c) = \Phi(x - \theta) is unique, so constructing an MVUE is just a matter of applying Rao-Blackwell and Lehmann-Scheffe.

However for our case, P(-c \le X \le c) is the same for \theta and -\theta. So it seems like the MVUE isn't unique. I'm wondering if you can make a decision rule like choosing one unbiased estimator when \theta \ge 0 and the other when \theta < 0, but now instead of two non-unique unbiased estimators, we have three. Any thoughts? Is a MVUE just not possible?

 

[Stephen Tashi]

Our task is to determine if P(-c \le X \le c) has a minimum variance unbiased estimator for a sample from a distribution that is N(\theta,1).

I assume this means that c is given and \theta is unknown. So you can't employ a rule that depends on knowing the sign of \theta.

 

[rayge]

What if we construct two MVUE's, one for P(X \le c), and one for P(X \le -c), and then subtract one from the other? It still seems like we have the same problem, where the MVUE is not one-to-one...

 

[Stephen Tashi]

However for our case, P(-c \le X \le c) is the same for \theta and -\theta.

There is ambiguity if you estimate P(-c \le X \e c) first and try to estimate \theta from that estimate. However the problem you stated doesn't insist we estimate \theta in that manner. Wouldn't the simplest try be to estimate \theta from the sample mean and then estimate P(-c \le X \le c) from that estimate?