Let x1, . . ,xn ∈ r be independent random variables with a common cdf f0. let fn be their ecdf and let f be any cdf. if f = fn, then l(f) < l(fn).

See the proof below.

Step-by-step explanation:

For this case we need to proof that: Let be independent random variables with a common CDF . Let be their ECDF and let F any CDF. If then

Proof

Let different values in the set {} and we can assume that represent the number of that are equal to .

We can define and assuming the probability .

For the case when for any then we have that the

And for the case when all and for at least one we know that for all the possible values . So then we can define the following ratio like this:

So then we have that:

And the log for a number is 0 or negative when the number is between 0 and 1, so then on this case we can ensure that

And with that we complete the proof.

step-by-step explanation:

a and d