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?

 

[blue_raver22]

I can provide a response to the above content by saying that the concept of MVUE (Minimum Variance Unbiased Estimator) is a statistical tool used to estimate a population parameter with minimum possible variance and without any bias. In the case of a two-sided interval of a normal distribution, the task is to determine if there exists an MVUE for the parameter \theta.

In the given scenario, we have a normal distribution with a mean of \theta and a variance of 1. The one-sided interval P(X \le c) = \Phi(x - \theta) is unique, and we can apply Rao-Blackwell and Lehmann-Scheffe to construct an MVUE.

However, when we consider the two-sided interval P(-c \le X \le c), we see that it is the same for \theta and -\theta. This leads us to believe that the MVUE may not be unique in this case. It is possible to have multiple unbiased estimators for the same parameter, and this can make it challenging to determine the MVUE.

One approach to dealing with this issue could be to use a decision rule, as mentioned in the content. This approach involves choosing one unbiased estimator when \theta \ge 0 and another when \theta < 0. However, this would result in three non-unique unbiased estimators, which may not be ideal.

In conclusion, it may be challenging to determine an MVUE for the parameter \theta in this scenario. It is possible that an MVUE does not exist in this case, or it may require further exploration and analysis to find a suitable solution. As a scientist, it is important to carefully consider all possible options and make informed decisions based on statistical principles and methods.