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order statistics (Definition)

Let $ X_1,\ldots,X_n$ be random variables with realizations in $ \mathbb{R}$. Given an outcome $ \omega$, order $ x_i=X_i(\omega)$ in non-decreasing order so that

$\displaystyle x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}.$
Note that $ x_{(1)}=\operatorname{min}(x_1,\ldots,x_n)$ and $ x_{(n)}=\operatorname{max}(x_1,\ldots,x_n)$. Then each $ X_{(i)}$, such that $ X_{(i)}(\omega)=x_{(i)}$, is a random variable. Statistics defined by $ X_{(1)},\ldots,X_{(n)}$ are called order statistics of $ X_1,\ldots,X_n$. If all the orderings are strict, then $ X_{(1)},\ldots,X_{(n)}$ are the order statistics of $ X_1,\ldots,X_n$. Furthermore, each $ X_{(i)}$ is called the $ i$th order statistic of $ X_1,\ldots,X_n$.

Remark. If $ X_1,\ldots,X_n$ are iid as $ X$ with probability density function $ f_X$ (assuming $ X$ is a continuous random variable), Let $ \textbf{Z}$ be the vector of the order statistics $ (X_{(1)},\ldots,X_{(n)})$ (with strict orderings), then one can show that the joint probability density function $ f_{\textbf{Z}}$ of the order statistics is:

$\displaystyle f_{\textbf{Z}}(\boldsymbol{z})=n!\prod_{i=1}^{n}f_X(z_i),$
where $ \boldsymbol{z}=(z_1,\ldots,z_n)$ and $ z_1<\cdots<z_n$.



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Cross-references: vector, continuous random variable, probability density function, iid, strict, orderings, statistics, order, outcome, random variables
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This is version 5 of order statistics, born on 2004-08-24, modified 2006-09-15.
Object id is 6111, canonical name is OrderStatistics.
Accessed 11281 times total.

Classification:
AMS MSC62G30 (Statistics :: Nonparametric inference :: Order statistics; empirical distribution functions)

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