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?

  • $\begingroup$ 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? $\endgroup$ – whuber Jun 24 '20 at 13:34
  • $\begingroup$ @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. $\endgroup$ – Andrey Gorbunov Jun 24 '20 at 14:20
  • $\begingroup$ 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. $\endgroup$ – whuber Jun 24 '20 at 15:49
  • $\begingroup$ @whuber, I've modified accordingly using $\mathbb{E}[x]$, which is a definite value, instead of $\bar{x}$. $\endgroup$ – Andrey Gorbunov Jun 24 '20 at 15:51
  • $\begingroup$ 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? $\endgroup$ – whuber Jun 24 '20 at 15:52

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