I have a random variable with an unknown distribution and I want to find its cumulative distribution function. I sample the distribution $N$ times, with $$X_1, \dots, X_N$$ being iid random variables. I then plot the empirical cumulative distribution function.

Using this article https://en.wikipedia.org/wiki/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality

Am I correct in assuming that if I let $N$ be such that $$N \geq \frac{1}{2\epsilon^2}\ln(\frac{2}{\alpha})$$ Then my empirical distribution function is within $\epsilon$ of the actual cumulative distribution function at all points with a $1-\alpha$ confidence?


1 Answer 1


Yes, if $N \geq \frac{1}{2\epsilon^2}\ln(\frac{2}{\alpha})$, then $$ \mathbb{P}\left(\sup_{x \in \mathbb{R}}|F_N(x) - F(x)| > \epsilon \right) \le 2 e^{-2N\epsilon^2} \le \alpha, $$ which is what you want.


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