Kolmogorov test

Kolmogorov (test)

One of the most important goodness-of-fit tests, together with the goodness-of-fit Chi-square test.

Given:

1) A sample of observations on a numerical variable x,

2) And a completely determined cumulative distribution function F(x),

the Kolmogorov test will test the H₀ hypothesis according to which the sample originates from the distribution described by F(x).

For that purpose, it calculates a quantity D, the "Kolmogorov statistic" from the sample. D is a measure of the departure of the empirical cumulated distribution function from F(x). The theoretical distribution of D when H₀ is true is known, which is a remarkable achievement. A large value of D is an indication that the sample distribution function departs substantially from F(x), and leads to rejecting H₀.

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The Kolmogorov-Smirnov test is a two-sample test, that tests the H₀ hypothesis according to which two samples (numerical values) originate from the same (undetermined) distribution function F(x). It is based on the same principle as the (one sample) Kolmogorov test.

The test first calculates F₁(x) and F₂(x) , the respective cumulated distribution functions of the two samples. A quantity D, that measures the discrepancy between these two functions, is then calculated. The theoretical distribution of D when H₀ is true is known. A large value of D is an indication that the two samples are too different for being reasonably believed to have been generated by the same underlying probability distribution, and leads to the rejection of H₀.