empirical distribution function

Let X1,,Xn be random variablesMathworldPlanetmath with realizations xi=Xi(ω), i=1,,n. The empirical distribution function Fn(x,ω) based on x1,,xn is


where χAi is the indicator functionPlanetmathPlanetmath (or characteristic functionMathworldPlanetmathPlanetmath) and Ai={(x,ω)xix}. Note that each indicator function is itself a random variable.

The empirical function can be alternatively and equivalently defined by using the order statisticsMathworldPlanetmath X(i) of Xi as:

Fn(x,ω)={0if x<x(1);1nif x(1)x<x(2)1k<2;2nif x(2)x<x(3)2k<3;inif x(i)x<x(i+1)ik<i+1;1if xx(n);

where x(i) is the realization of the random variable X(i) with outcome ω.

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