# Different ways of writing the Kolmogorov-Smirnov test statistic

I have a question on the Kolmogorov-Smirnov test.

This is what I know about the test.

Consider the sequence of i.i.d. random variables $\{Z_i\}_{i=1}^n$ and let $\{z_i\}_{i=1}^n$ be one possible realisation of the sequence.

We want to test $$H_0: Z_1\sim F \text{ vs } H_1: Z_1\nsim F$$ where $F:\mathbb{R}\rightarrow [0,1]$ is the reference CDF for $Z_1$.

The empirical cumulative distribution function of $Z_1$ is $F_n:\mathbb{R}\rightarrow [0,1]$ prescribed by $$F_n(z):=\frac{1}{n}\sum_{i=1}^n1(z_i\leq z)$$ $\forall z \in \mathbb{R}$.

The KS test statistic is $$KS_n:=\sup_{z\in \mathbb{R}}|F_n(z)-F(z)|$$

We reject $H_0$ if $\sqrt{n}KS_n>c^{KS}_{1-\alpha}$.

I found this paper about the two-stage delta correction of the KS test which explains the classical KS test statistic in another way, i.e., it introduces as test statistic $$\tilde{KS}_n:=\max_{1\leq i\leq n} |\frac{i-\frac{1}{2}}{n}-F_i|$$ where $\frac{i-\frac{1}{2}}{n}$ is the plotting position.

Question: how is $KS_n=\tilde{KS}_n$?

• $F_i = ?$ Or What is $F_i$? – user158565 Jun 3 '17 at 0:03
• I am not even sure about what is indicating $i$ here.. – user3285148 Jun 3 '17 at 0:38
• I try to find $i$ and $F_i$, but your link does not work for me, because I do not want to pay money. – user158565 Jun 3 '17 at 3:05