empirical distribution function
Let X1,…,Xn be random variables with realizations xi=Xi(ω)∈ℝ, i=1,…,n. The empirical distribution function Fn(x,ω) based on x1,…,xn is
Fn(x,ω)=1nn∑i=1χAi(x,ω), |
where χAi is the indicator function (or characteristic function
) and Ai={(x,ω)∣xi≤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 Xi as:
Fn(x,ω)={0if x<x(1);1nif x(1)≤x<x(2), 1≤k<2;2nif x(2)≤x<x(3), 2≤k<3;⋮inif x(i)≤x<x(i+1), i≤k<i+1;⋮1if x≥x(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 |