Let say I have the following parameter to estimate:
$$ \theta = \frac{1}{\mu^2 - 1} \ .$$
The observed measurement is $x \sim \mathcal{N}(\mu,\sigma)$, $\mu^2\neq1$. The mean $\mu$ is unknown. There are two cases I am interested in:
- The value $\sigma$ is known
- The value of $\frac{\sigma}{\mu}$ is known
What is a minimum mean square error (MMSE) estimator $\hat{\theta}(x) = \text{argmin}_{\theta_0(x)} \mathbb{E}[(\theta_0(x) - \theta)^2] $ for the given function? Will it be unbiased?
(I have asked the same question on Cross Validated 3 days ago.)