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Illustration of the Kolmogorov–Smirnov statistic. The red line is a model CDF, the blue line is an empirical CDF, and the black arrow is the KS statistic.. In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions.
Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров, IPA: [ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf] ⓘ, 25 April 1903 – 20 October 1987) [4] [5] was a Soviet mathematician who played a central role in the creation of modern probability theory.
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. [1] These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. [2] There are several other (equivalent) approaches to formalising ...
Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for the average to converge almost surely on something (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that). [22]
Kolmogorov's theorem is any of several different results by Andrey Kolmogorov: In statistics. Kolmogorov–Smirnov test; In probability theory. Hahn–Kolmogorov theorem; Kolmogorov extension theorem; Kolmogorov continuity theorem; Kolmogorov's three-series theorem; Kolmogorov's zero–one law; Chapman–Kolmogorov equations; Kolmogorov ...
William Feller, in 1949, used the names "forward equation" and "backward equation" for his more general version of the Kolmogorov's pair, in both jump and diffusion processes. [1] Much later, in 1956, he referred to the equations for the jump process as "Kolmogorov forward equations" and "Kolmogorov backward equations". [3]
The Kolmogorov structure function precisely quantifies the goodness-of-fit of an individual model with respect to individual data. The Kolmogorov structure function is used in the algorithmic information theory, also known as the theory of Kolmogorov complexity, for describing the structure of a string by use of models of increasing complexity.
In probability theory, Kolmogorov's Three-Series Theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions.