PlanetMath: empirical distribution function

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empirical distribution function

(Definition)

Let $X_1,\ldots,X_n$ be random variables with realizations $x_i=X_i(\omega)\in\mathbb{R}$ , $i=1,\ldots,n$ . The empirical distribution function $F_n(x,\omega)$ based on $x_1,\ldots,x_n$ is

$\displaystyle F_n(x,\omega)=\frac{1}{n}\sum_{i=1}^{n}\chi_{A_i}(x,\omega),$

where $\chi_{A_i}$ is the indicator function (or characteristic function) and $A_i=\lbrace(x,\omega)\mid x_i\leq x \rbrace$ . Note that each indicator function is itself a random variable.

The empirical function can be alternatively and equivalently defined by using the order statistics $X_{(i)}$ of as:

$\begin{displaymath} F_n(x,\omega)= \begin{cases} 0 & \text{if $x<x_{(1)}$;}\ \... ...<i+1$;}\ \vdots\ 1 & \text{if $x\geq x_{(n)}$;} \end{cases}\end{displaymath}$

where $x_{(i)}$ is the realization of the random variable $X_{(i)}$ with outcome $\omega$ .

"empirical distribution function" is owned by CWoo.

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Cross-references: outcome, order statistics, function, characteristic function, indicator function, random variables
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This is version 4 of empirical distribution function, born on 2004-08-24, modified 2007-12-15.
Object id is 6110, canonical name is EmpiricalDistributionFunction.
Accessed 7139 times total.

Classification:

AMS MSC:

62G30 (Statistics :: Nonparametric inference :: Order statistics; empirical distribution functions)

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