EmpiricalDistributionFunction 2004-08-24 19:34:45 2007-12-15 11:00:37 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $X_1,\ldots,X_n$ be random variables with realizations $x_i=X_i(\omega)\in\mathbb{R}$, $i=1,\ldots,n$. The \emph{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 $A_i=\lbrace(x,\omega)\mid x_i\leq x \rbrace$. Note that each indicator function is itself a random variable. \par 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 $\omega$.