# Nonlinear minimum Mean Square Error estimation

Let say I have the following parameter to estimate:

$$\theta = \frac{1}{\mu^2 - 1} \ .$$

The observed measurement is $$x \sim \mathcal{N}(\mu,\sigma)$$. The mean $$\mu$$ is unknown. There are two cases I am interested in:

1. The value $$\sigma$$ is known
2. 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?

• This question doesn't make much sense when the words are interpreted in standard ways. A "parameter" cannot be a function of data like $\bar x.$ Could you clarify what kind of object $\theta_0$ is? – whuber Jun 24 '20 at 13:34
• @whuber, $\theta$ is an estimand (en.wikipedia.org/wiki/Estimand) and $\theta_0$ is the value when measurement $x$ doesn't have noise $\epsilon$. If it makes sense now, please edit the question using the correct terminology. – Andrey Gorbunov Jun 24 '20 at 14:20
• It doesn't make sense yet because an estimand, by definition, cannot be a statistic such as $\bar x.$ The target of an estimator (which is what $\theta$ is intended to be) must be a definite property of the underlying distribution from which the data are sampled. You are using the symbol "$x$" in two distinct senses, both as a random value and as a parameter. – whuber Jun 24 '20 at 15:49
• @whuber, I've modified accordingly using $\mathbb{E}[x]$, which is a definite value, instead of $\bar{x}$. – Andrey Gorbunov Jun 24 '20 at 15:51
• Are you perhaps trying to ask for a minimum-variance estimator of $1/(\mu^2-1)$ based on an observation from $\mathcal{N}(\mu,\sigma)$? If so, is $\sigma$ known or not? – whuber Jun 24 '20 at 15:52