# What exactly is an estimator? A function or the value of a function?

The formal definition of an estimator is :

The function of $$X_1, X_2,\ldots , X_n$$, that is, the statistic $$T(X_1, X_2, \ldots, X_n)$$, used to estimate $$\theta$$ is called a point estimator of $$\theta$$.

1. the first sentence just says that an estimator is a function $$T$$ of some random variable.
2. the second sentence says that an estimator is the value $$T(X_1, X_2, \ldots, X_n)$$. From my understanding, this is not a function. This is the value of a function $$T$$ applied to some random variables.

For example: $$Y=T(X_1, X_2, \dots, X_n)$$

the definition just means that an estimator is both the function $$T$$ and the function value $$Y$$.

In some Textbook, I have seen $$T=T(X_1, X_2, \ldots, X_n)$$ to mean an estimator or a statistic.

I think this is overload because how can a thing being a function and the result at the same time?

• When I say $f=f(x)$, usually, I am not saying that f is the result of $f(x)$ after evaluating the function at $x$. What I am saying is just that both $f$ and $f(x)$ denote the same thing, namely, a function that takes as input one argument. Is that what you are asking? Jul 17, 2019 at 3:53
• Note that a "statistic", strictly speaking, is a function, not the value of a function. So an estimator is a function, and the value of the estimator when applied to a given data set is an estimate. But informally we often use the same term for both - so "standard deviation" can mean the estimator or the estimate. Jul 17, 2019 at 10:06

$$T = T(X_1,X_2,..X_n)$$ is the Estimator which is the function. And the value it evaluates to, in a particular experiment is known as the Estimate.