I am studying random signals and noise (a course for EE students, but mathematical and formal), and have a question about the definition of an estimator (in the context of estimating a random variable from a random variable):
Given that $X$ and $Y$ are random variables, we want to estimate $X$ from $Y$.
Now the inconsistency between different sources begins:
One professor of the course defined an estimator as a function of the observable data $Y$ - i.e., an estimator is the random variable $g(Y)$ (that is to say, he defined an estimator as a composition of a function $g: \mathbb{R} \to \mathbb{R}$ over the random variable $Y: \Omega \to \mathbb{R}$ - $\hat{X} = g \circ Y : \Omega \to \mathbb{R}$).
Another professor defined an estimator as a function $g:\mathbb{R} \to \mathbb{R}$, which is a different definition and meaning from the first professor.
I do understand that this is nitpicky, but I have time, so I wanted to verify this with you.
Thank you in advance!
