The empirical distribution function (EDF) of a finite sequence of realizations of a random variable lies at the heart of statistical inference. We will first introduce the concept of estimation and then define the EDF.
Consider a random variable X with distribution function 
. We call 
a random sample from if, for all i, 
and the 
are
independent.
Let 
be a property of , e.g. a parameter in the model describing X such as the
expectation, the variance or a general statement like the probability that X will assume a
value less than t, 
.
Estimators are used to measure properties of distribution functions. In the field of statistical
inference they are used to construct models from empirical data. We have already
commented on the principal difficulties arising from such a setup: the estimations cannot be
falsified. However, they are a very important tool for both practical and theoretical statistics.
If an estimator is calculated from some k experiments yielding the random numbers
, the resulting number is a random number itself, i.e. a realization
of the random variable 
.
The above form of estimators is called 'point estimate' since it will yield a single real
value. There also exist so-called 'interval estimates' that yield intervals containing the
parameter with a given probability. We will make use of an estimator for the whole
distribution function , however. Such an estimator needs to be a function itself. We
therefore define the empirical distribution function.
The strong law of large numbers (S.L.L.N.) immediately gives pointwise convergence if the
random sample is drawn such
that the are independent:
The theorem of Glivenko-Cantelli (Theorem
20.6. in
[3]) even proves a uniform convergence in t:
The empirical distribution thus serves as an estimator for the distribution function of a
random
variable. It can easily be extended to the multivariate case by defining
where
and s is the dimension of the vector 
.
Note that the independence of the random variables is essential to get the
desired
limiting behavior of the EDF. It can easily be seen that the validity of the Glivenko-Cantelli
theorem remains valid if we substitute any measurable function g(X)
for X and consider the
distance of
the EDF 
and the distribution function 
. This
property
shows again the great difference between random variables and random numbers: if we fix a
sequence 
of random numbers, we could get the right limiting behavior, e.g.
convergence, for one function 
and a wrong behavior for another function 
!
This is due to the fact that our sequence may be contained in the set of measure zero for
which the
Glivenko-Cantelli theorem does not claim convergence for . This set can differ from
the according set for the function .
The term EDF denotes a random function, i.e. an infinite-dimensional random vector, as well as a concrete realization of this vector, which results from substituting random numbers for the random sample. From now on we will always refer to the second meaning of EDF, thus:
