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# empirical distribution function

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

$F_{n}(x,\omega)=\frac{1}{n}\sum_{{i=1}}^{{n}}\chi_{{A_{i}}}(x,\omega),$ |

where $\chi_{{A_{i}}}$ is the indicator function (or characteristic function) and $A_{i}=\{(x,\omega)\mid x_{i}\leq x\}$. 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 $X_{i}$ as:

$F_{n}(x,\omega)=\begin{cases}0&\text{if $x<x_{{(1)}}$;}\\ \frac{1}{n}&\text{if $x_{{(1)}}\leq x<x_{{(2)}}$, $1\leq k<2$;}\\ \frac{2}{n}&\text{if $x_{{(2)}}\leq x<x_{{(3)}}$, $2\leq k<3$;}\\ \vdots\\ \frac{i}{n}&\text{if $x_{{(i)}}\leq x<x_{{(i+1)}}$, $i\leq k<i+1$;}\\ \vdots\\ 1&\text{if $x\geq x_{{(n)}}$;}\end{cases}$ |

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

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