Q:

Which is a statistic, $T(X)$ or $T$?

This is a terminology question.
Is $T(X)$ or $T$ a statistic? $X$ is a random variable, and $T$ is a measurable mapping.
I ask this because, I want to know how to say the distribution of $T(X)$ completely in words.
$T(X)$ is the composition of  $T$ and the random variable $X$, and i call the distribution of $T(X)$  the distribution of the composition of $T$ and random variable $X$. But can we simplify the term to be the distribution of the statistic? That leads to whether  $T$ or $T(X)$ should be called a statistic. thanks.

A:

Strictly speaking, a statistic is a function of the sample. $T(X)$ emphasizes that fact. But often, once the understanding is established that $T(X)$ is a random variable, we sometimes write $T$ and just view it as a single draw from its sampling distribution.

