Why is an estimator considered a random variable?

Question

My understanding of what an estimator and an estimate is: Estimator: A rule to calculate an estimate Estimate: The value calculated from a set of data based on the estimator

Between these two terms, if I am asked to point out the random variable, I would say the estimate is the random variable since it's value will change randomly based on the samples in the dataset. But the answer I was given is that the Estimator is the random variable and the estimate is not a random variable. Why is that ?

Answer 1

Somewhat loosely -- I have a coin in front of me. The value of the next toss of the coin (let's take {Head=1, Tail=0} say) is a random variable.

It has some probability of taking the value $1$ ($\frac12$ if the experiment is "fair").

But once I have tossed it and observed the outcome, it's an observation, and that observation doesn't vary, I know what it is.

Consider now I will toss the coin twice ($X_1, X_2$). Both of these are random variables and so is their sum (the total number of heads in two tosses). So is their average (the proportion of head in two tosses) and their difference, and so forth.

That is, functions of random variables are in turn random variables.

So an estimator -- which is a function of random variables -- is itself a random variable.

But once you observe that random variable -- like when you observe a coin toss or any other random variable -- the observed value is just a number. It doesn't vary -- you know what it is. So an estimate -- the value you have calculated based on a sample is an observation on a random variable (the estimator) rather than a random variable itself.

Answer 2

My understandings:

1. An estimator is not only a function, which input is some random variable and output another random variable, but also a random variable, which is just the output of the function. Something like $y=y(x)$, when we talk about $y$, we mean both the function $y()$, and the result $y$.
2. Example:an estimator $\overline X=\mu(X_1,X_2,X_3)=\frac{X_1+X_2+X_3}{3}$, we mean both $\mu()$ ,which is a function,and its result $\overline X$,which is random variable.
3. The difference between estimator and estimate is about before observing or after observing.
4. Actually, similar to an estimator, an estimate is both a function and a value(the function output) too. But the estimate is in the context of after observing, and by contrast, the estimator is in the context of before observing.

A picture illustrates the idea above:

I have researched this question during my weekend, after reading lots of material from the internet, i am still confused. Although I am not completely sure that my answer is right, it seems like, to me, it's the only way to let everything make sense.

Answer 3

Let $\left(\mathcal X, \mathcal A, \left(\mathbb P_\vartheta\right)_{\vartheta \in \Theta}\right)$ be a statistical model consisting of

• a set $\mathcal X$ (the sample space),
• a $\sigma$-algebra $\mathcal A$ on $\mathcal X$,
• a family of probability measures $\left(\mathbb P_\vartheta\right)_{\vartheta \in \Theta}$ on $\mathcal A$ with
• index set $\Theta$ (the parameter space) of cardinality bigger than one.

Let $(\Sigma, \mathcal S)$ be another event space, i.e., a measurable space.

Let $\tau: \Theta \to \Sigma$ be any function.

Then, an estimator $T$ of $\tau$ is defined to be any random variable from $(\mathcal X, \mathcal A)$ to $(\Sigma, \mathcal S)$.

In this answer I describe the relation between estimands, estimators, and estimates.

---

Reference

Georgii, H.-O. (2013). Stochastics: Introduction to probability and Statistics (E. Baake & M. Ortgiese, Trans.). Walter de Gruyter.