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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.

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  • $\begingroup$ If $F_X^{\,}$ is continuous then $F_X^{\,}(X_{(i)})$ has a Beta distribution with parameters $i$ and $n+1-i$ $\endgroup$ – Henry Sep 13 '17 at 11:45
  • $\begingroup$ I do not want to know how these are distributed, I want to know what they represent in the first place. $\endgroup$ – M. Van Sep 13 '17 at 11:45
  • $\begingroup$ It may seem circular, but $F_X^{\,}(X_{(i)})$ is simply the application of a function $F_X^{\,}(x)$ to a random variable $X_{(i)}$, and so it represents the application of the cumulative distribution function for $X$ to the $i$th order statistic of a random sample size $n$ from the distribution. In a hand-waving way, it is a measure of how far along the original distribution the $i$th order statistic appears; for continuous distributions it does not depend on the the particular distribution $\endgroup$ – Henry Sep 13 '17 at 11:57
  • $\begingroup$ If $Y$ is a random variable so is $F(Y)$ by the recipe $\omega\mapsto F(Y(\omega))$. In the case at hand $Y=X_{(i)}$ which has a somewhat complicated formula in terms of the $X_i(\omega)$ $\endgroup$ – kimchi lover Sep 13 '17 at 12:01
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    $\begingroup$ @Henry Thank you! So in pedantic notation, $F(X_{(i)}=F \circ X_{(i)}$. This makes sense because $X_{(i)}: \Omega \rightarrow \mathbb{R}$ and $F:\mathbb{R} \rightarrow [0,1]$. $\endgroup$ – M. Van Sep 13 '17 at 12:06
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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.

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