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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?

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  • $\begingroup$ 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? $\endgroup$
    – evaristegd
    Jul 17, 2019 at 3:53
  • $\begingroup$ 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. $\endgroup$
    – gandalf61
    Jul 17, 2019 at 10:06

1 Answer 1

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

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