Order Statistic

Topics: Statistics

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(definition)

Let $X_1,\dots ,X_n$ be a random sample. The variables:

$$\begin{array}{rl}{X}_{(1)}& =\min \left\{X_1,\dots ,X_n\right\}\\ X_{(2)}& =\text{2nd }\min \left\{X_1,\dots ,X_n\right\}\\ & ⋮\\ X_{(n)}& =\max \left\{X_1,\dots ,X_n\right\}\end{array}$$

…are called order statistics, where $X_{(i)}$ is the order statistic of order $i=1,\dots ,n$.

Note that these are just random variables defined as so, nothing more. Once we observe the sample and obtain every realisation $x_i$, then the order relation of the realisations is used to obtain, in turn, the realisations $x_{(i)}$ of the order statistics.

Properties §

(proposition)

Concerning the density functions of order statistics:

1. $f_{X_{(1)}}(x)=nf(x)[1-F(x){]}^{n-1}$
2. $f_{X_{(n)}}(x)=nf(x)[F(x){]}^{n-1}$
3. $f_{X_{(i)}}(x)=\left(\genfrac{}{}{0ex}{}{n}{i}\right)if(x)[F(x){]}^{i-1}[1-F(x){]}^{n-i}$

…where $f(x)$ and $F(x)$ are the density and distribution functions of every $X_i$ (remember that since they’re a random sample, they are iid).

What really are these?

Do note that these are not the density functions of the random variable that corresponds to the $(i)$th random variable; that would just be $f(x)$. Rather, they are the density function of the random variable that is defined as:

$$X_i=i\text{th min}\left\{X_1,\dots ,X_n\right\}$$

To better understand what they are, know that (distribution function):

$$f_{X_{(i)}}(x)=\frac{d}{dx}\left[F_{X_{(i)}}(x)\right]=\frac{d}{dx}\left[\mathbb{P}(X_{(i)}\le x)\right]$$

…where $\mathbb{P}(X_{(i)}\le x)$ means $\mathbb{P}(\text{at least }i\text{ of the }n\text{ total }{X}_i\text{ are}\le x)$.