I have followed the Cramér-Rao lower bound (CRLB) derivation, and I couldn't figure out why –
If $f(x; \theta)$ be a probability density with continuous parameter $\theta$, and $X_1, \dots, X_n$ be independent random variables with density $f(x; \theta)$, and $\Theta(X_1, \dots ,X_n)$ be an unbiased estimator of $\theta$.
Why does the estimator, $\Theta$, is independent of $\theta$ (the param to be estimated)?
$\Theta$ is a function of $X_1, \dots, X_n$, and in the pdf of each one of them $\theta$ appears as a param. Doesn't it imply that $\Theta$ is also dependent on $\theta$?
Best Answer
I'm not too fond of the use of the word "independent" in this context. What the author meant is that the estimator $\Theta = \Theta (X_1, \ldots , X_n)$ is a function only of the data, and not of the parameter $\theta$ to be estimated. Intuitively, we want to be able to observe $\Theta$; since $\theta$ is an unobservable (yet, estimable) quantity, we don't want to incorporate it in the definition of $\Theta$.
The fact that $\theta$ appears in the density of $X_i$ will mean that it will appear in the density of $\Theta$, but this does not mean that $\Theta$ is a function of $\theta$.