# Cramér-Rao lower bound with an uncertain observable

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?

• What about if $X$ is normally distributed? (I updated the question to signify that) Apr 17 '20 at 11:11

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.