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 $\omega$ .

Title	empirical distribution function
Canonical name	EmpiricalDistributionFunction
Date of creation	2013-03-22 14:33:27
Last modified on	2013-03-22 14:33:27
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 62G30