Notation for estimator in statistics

Question

I'm trying to understand the estimator notation $\hat\theta$ and what is correct use.

If $\hat\theta = \hat\mu$, is $\hat\mu(X) = \overline X$?

Is $\hat p(X,n) = \dfrac{X}{n}$?

If not, what is correct notation for these?

Answer 1

It is best to start by naming the distribution. If $X_1, \dots, X_n$ are iid $N(\mu, \sigma_0^2),$ with $\sigma_0$ known and $\mu$ to be estimated, then one might consider several estimators of $\mu.$ If $\bar X$ is one of them, one might write $\hat \mu = \bar X.$ If the median is another, then that might be denoted by $\tilde \mu.$ Some authors like to reserve the 'hat' notation $\hat{}$ for maximum likelihood estimators, or ones with some other optimal properties.

If a large number of different estimators for the same population parameter is under consideration, they might just be called something like $T_1,$ $T_2,$ and so on, without resorting to 'decorations' on the letter denoting the parameter.

Attempting more compact notation, some authors invent their own shorthand, which should be clearly explained at its first usage. I have seen a vast variety of different notations by different authors--some clearly explained and some not. Perhaps a failure of clear explanation has given rise to your question.