In statistics, it is very important to differentiate between thesethe following three concepts which are often confused and mixed by students.
Usually, books denote by $\theta$ an unknown parameter. Then, we wish to estimate it. We use an estimator which books usually denote by $\widehat{\theta}$. The estimator is a random variable! Usually we seek $E[\widehat{\theta}]=\theta$ and so on and on, anyways. An estimate is the value we obtain by sampling and inserting our values in our estimator.
A classical example is: parameter: population mean $\mu$. Estimator $\overline{X}=\frac{1}{n}\sum_{i=1}^{n} X_i$ based on a priori observations $X_1,\dots,X_n$. And then, we sample observations $x_1,\dots,x_n$ and compute $\overline{x}=\frac{1}{n}\sum_{i=1}^{n} x_i$. Obs: $\mu$ is an unknown number. $\overline{X}$ is a random variable, and $\overline{x}$ is a number!
- Parameter: population mean $\mu$.
- Estimator $\overline{X}=\frac{1}{n}\sum_{i=1}^{n} X_i$ based on a priori observations $X_1,\dots,X_n$. And then, we sample observations $x_1,\dots,x_n$ and compute $\overline{x}=\frac{1}{n}\sum_{i=1}^{n} x_i$.
- Obs: $\mu$ is an unknown number. $\overline{X}$ is a random variable, and $\overline{x}$ is a number!
With concrete latin letters it seems easy to stress this fact, but when we use $\theta$ and $\widehat{\theta}$ (the classical hat notation for estimator) I do not know how to stress this fact. I do not know how to differentiate between $\widehat{\theta}$ and a specific observed $\widehat{\theta}$.
Some books propose: $\widehat{\theta}_{obs}$ which I do not like for instance if we talk about two population proportions $p_1$ and $p_2$ and their estimators $\widehat{p}_1$ and $\widehat{p}_2$. Because then it would look like $\widehat{p}_{1,obs}$ which is not aesthetic.
What solutions do you propose? Has anyone seen a nice notation for this?
Thank you very much!
