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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.
In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large numbers .
1: Normality test: sample size between 3 and 5000 [16] Kolmogorov–Smirnov test: interval: 1: Normality test: distribution parameters known [16] Shapiro-Francia test: interval: univariate: 1: Normality test: Simpliplification of Shapiro–Wilk test Lilliefors test: interval: 1: Normality test
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 ...
Confidence bands can be constructed around estimates of the empirical distribution function.Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods.
Lilliefors test is a normality test based on the Kolmogorov–Smirnov test.It is used to test the null hypothesis that data come from a normally distributed population, when the null hypothesis does not specify which normal distribution; i.e., it does not specify the expected value and variance of the distribution. [1]
Kolmogorov's zero–one law asserts that, if the F n are stochastically independent, then for any event (()), one has either P(E) = 0 or P(E)=1. The statement of the law in terms of random variables is obtained from the latter by taking each F n to be the σ-algebra generated by the random variable X n .
Kuiper's test is closely related to the better-known Kolmogorov–Smirnov test (or K-S test as it is often called). As with the K-S test, the discrepancy statistics D + and D − represent the absolute sizes of the most positive and most negative differences between the two cumulative distribution functions that are being compared.