Cramér-Rao lower bound with an uncertain observable

Question

Let's say I have an observable $X$ that depends on a parameter $\theta$, and that I can find an expression for the Cramér-Rao lower bound for estimating $\theta$, as a function of $X$: $\sigma_{CRB}(X)$, by assuming I can exactly measure $X$. Now let's say I cannot measure $X$ exactly, and instead it is normally distributed. Now it seems to me there should be a way to work this into the "correct" Cramér-Rao bound, is this correct and how would it be done?

Answer 1

Recall the Fréchet-Darmois-Cramér-Rao lower bound conditions [reproduced from Wikipedia]:

Suppose $\theta$ is an unknown deterministic parameter which is to be estimated from $n$ independent observations (measurements) of $X$, each distributed according to some probability density function ${\displaystyle f(x;\theta )}$. ). The variance of any unbiased estimator ${\displaystyle {\hat {\theta }}_n}$ of ${\displaystyle \theta }$ is then bounded by the reciprocal of the Fisher information ${\displaystyle I(\theta )}$ associated with $n$ iid observations: $${\displaystyle \operatorname {var} ({\hat {\theta }_n})\geq {\frac {1}{n I(\theta )}}}$$

Therefore, to apply in this framework when observing $y_i=X_i+\epsilon_i$,

1. There must exist an unbiased estimator based on the $Y_i$'s, which is not an immediate consequence of an existing unbiased estimator based on the $X_i$'s, unless the estimator is linear in the $X_i$'s, like $\hat{\theta}_n=\bar{X}_n$;
2. The Fisher information is the Fisher information associated with the distribution of $Y$, rather than the Fisher information associated with the distribution of $X$. Deriving one from the other is rarely feasible.