Kolmogorov Smirnov Test

Should be used if data sequence is continous.

What can we test with this test?

• GOF between two samples
• If one of two samples is stochastically bigger or smaller than the other sample
• GOF for one sample to a certain CDF
• Using Lilliefors Test of Normality we can test if a sample comes from a class of normal distributions

GOF

The Null Hypothesis is $H_0:F=G$

The Test Statistic is given by the theorem of Glivenko-Cantelli with

$D_n\left(X_1,\dots ,X_n\right):={\sup }_{t\in \mathbb{R}}\left|F_n\left(t;X_1,\dots ,X_n\right)-F(t)\right|$

Given $H_0$ the test statistic should be small.

We can implement a supremum function in R like this:

sup = function(x, F){
  fn = ecdf(x)
  xi = c(knots(fn),Inf)
  eta = c(-Inf, knots(fn))
  max(abs(fn(xi) - F(xi)), abs(fn(eta) - F(xi)))
}

where knots gives the jump positions of a function. One can then use the function to get the supremum of the differences between the ECDF $x$ and a CDF like this:

sup(x, function(t) punif(t))

We have for the slightly changed Test Statistic (multiplied by $\sqrt{n}$) : ${\lim }_{n\to \infty }\mathbb{P}\left(\sqrt{n}{D}_n\left(X_1,\dots ,X_n\right)\le t\right)=K(t)$ where $K(t)$ is the Kolmogorov Distribution.

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We can also test the Null Hypothesis $H_0:F\le G$ with the Test Statistic $D_n^{+}:={\sup }_{t\in \mathbb{R}}\left(F_n\left(t;X_1,\dots ,X_n\right)-F(t)\right)$ for which we can simulate the Distribution with any continous distribution as it is independent on the underlying distribution if it is continous.

Now for large $n$ the distributon will have an explicit approximate form, a variation of the Kolmogorov Distribution: $K^{+}(t)=\left(1-e^{-2t^2}\right)1_{(0,\infty )}(t)$

How to perform the test

• Formulate Null Hypothesis
• Fix probability of Type 1 Error
• Determine $1-\alpha$ Quantile of the Kolmogorov Distribution
• Perform experiment and calculate Test Statistic
• accept $H_0$ if $D_n\left(x_1,\dots ,x_n\right)\le \frac{K_{1-\alpha }}{\sqrt{n}}$

Test of Equality of two continous distributions

$H_0:F=G$

Test Statistic

In R

Test if equal

x = rnorm(100)
y = rnorm(100, 0.2, 1.1)
 
ks.test(x=x, y=y, alternative="two.sided")

Test if equal to some CDF

x = rnorm(100)
 
ks.test(x=x, "norm", alternative="two.sided")

Test if less or greater:

x = rnorm(100)
y = rnorm(100, 0, -1)
 
ks.test(x=x, y=y, alternative="l")
ks.test(x=x, y=y, alternative="g")