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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 $KS_n=\tilde{KS}_n$?

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

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