# 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

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

Title empirical distribution function EmpiricalDistributionFunction 2013-03-22 14:33:27 2013-03-22 14:33:27 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 62G30