44 lines
726 B
Markdown
44 lines
726 B
Markdown
# Kolmogorov - Smirnov test
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## KS
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Quantify distance between expected and observed CDF
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. . .
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KS statistic:
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$$
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D_N = \text{sup}_x |F_N(x) - F(x)|
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$$
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- $F(x)$ is the expected CDF
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- $F_N(x)$ is the empirical CDF of $N$ sampled points
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- sort points in ascending order
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- number of points preceding the point normalized by $N$
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## KS
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$H_0$: points sampled according to $F(x)$
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. . .
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If $H_0$ is true:
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- $\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K$
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Kolmogorov distribution with CDF:
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$$
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P(K \leqslant K_0) = 1 - p = \frac{\sqrt{2 \pi}}{K_0}
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\sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
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$$
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. . .
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a $p$-value can be computed
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- At 95% confidence level, $H_0$ cannot be disproved if $p > 0.05$
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