Kolmogorov - Smirnov test

KS

Quantify distance between expected and observed CDF

. . .

KS statistic:


  D_N = \text{sup}_x |F_N(x) - F(x)|

F(x) is the expected CDF
F_N(x) is the empirical CDF of N sampled points
- sort points in ascending order
- number of points preceding the point normalized by N

H_0: points sampled according to F(x)

. . .

If H_0 is true:

Kolmogorov distribution with CDF:


  P(K \leqslant K_0) = 1 - p = \frac{\sqrt{2 \pi}}{K_0}
  \sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}

. . .

a $p$-value can be computed