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 ?
Best Answer
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.