Definition of $F(X_{(i)})$ as random variable

Question

A very basic question (I am new to statistics). In my book on statistics, they define the order statistics of a 'sample' (which I just see as a sequence $X_1,..,X_n$ of random variables, correct me if I'm wrong). Just a line further they make a statement about the expectation random variable $F(X_{(i)})$, if the $X_i$ are distributed with cdf $F$. I do not understand what this random variable is. For example, if $X_{(i)}=1$, what is $F(X_{(i)})$? Or more formally, if $\Omega$ is the sample space, what is $F(X_{(i)})(\omega)$ for $\omega \in \Omega$? Thanks in advance.

Answer 1

I think there might be a misunderstanding with the notation. Statistics textbooks usually denote a sample as $X_{1}, X_{2}, \ldots, X_{n}$. The ordered sample (usually in ascending order) is usually denoted as $X_{(1)}, X_{(2)}, \ldots, X_{(n)}$.

When you write $F(X)$, you are referring to the distribution of the random variable. But when you write $F(X_{(n)})$, you are referring to the distribution of the maximum observed value in your sample.

A classic textbook problem on the the distribution of order statistics is to find the distribution of the sample maximum, taken from a uniform distribution. Here is an example.

I hope this is what you were looking for.