The empirical distribution function of a sample 
, ..., 
is the distribution
function
![F_n(x)=1/nsum_(i=1)^n1_((-infty,x])(X_i)=1/n#{i:X_i<=x},](/images/equations/EmpiricalDistributionFunction/NumberedEquation1.svg) |
where 
is the indicator function of a set 
. It is a right-continuous nondecreasing step function whose
jump at an observed value equals that value's multiplicity divided by 
. Equivalently, it is the distribution
function of the discrete probability measure
that assigns mass 
to each observation.
If the observations are independent and identically distributed with distribution
function 
, the following uniform convergence holds:
 |
Thus 
converges uniformly to 
almost surely. Empirical
distribution functions are used to define sample quantiles and construct distribution-free
tests. In the nonparametric bootstrap method,
observations are resampled with replacement directly from 
. Unlike a parametric bootstrap,
no family of probability distributions is fitted first.
