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I have a random iid sample Xi, i=1, ..., n

The empirical cdf of the sample at poin s is

[tex]\hat{F}\left(s\right)=n^{-1}\sum_{i=1}^{n}{\textbf{1}_{\left(-\infty, s\right)}\left(x_{i}\right)}[/tex]

Clearly [tex]\hat{F}\left(s\right)[/tex] is binomially distributed with parameters n and p=F(t) (true cdf).

Now I need to find the covariance between [tex]\hat{F}\left(s\right)[/tex] and [tex]\hat{F}\left(t\right)[/tex] for s<t.

I know that the result is [tex]n^{-1}F\left(s\right)\left(1-F\left(t\right)\right)[/tex]

Any help is very appreciated...

Thanks!!

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# Cov between empirical cdf

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