Kolmogorov distance

E174592

metric on probability distributions probability metric statistical distance

Kolmogorov distance is a statistical metric that measures the maximum difference between two cumulative distribution functions, commonly used to quantify convergence in distribution and in goodness-of-fit tests.

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All labels observed (5)

Label	Occurrences
Kolmogorov–Smirnov test	3
Kolmogorov distance canonical	1
Kolmogorov metric	1
Kolmogorov–Smirnov metric	1
Kolmogorov–Smirnov statistic	1

How this entity was disambiguated

This entity first appeared as the object of triple T1535776 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Kolmogorov distance
Context triple: [Berry–Esseen theorem, typicalMetric, Kolmogorov distance]

A. Kullback–Leibler divergence
Kullback–Leibler divergence is a fundamental information-theoretic measure that quantifies how one probability distribution differs from a reference distribution.
B. Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
C. Rényi divergence
Rényi divergence is a family of information-theoretic measures that generalize Kullback–Leibler divergence to quantify the dissimilarity between probability distributions, parameterized by an order α.
D. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
E. Rényi entropy
Rényi entropy is a generalized measure of information and uncertainty that extends Shannon entropy by introducing a tunable order parameter to emphasize different aspects of a probability distribution.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Kolmogorov distance
Target entity description: Kolmogorov distance is a statistical metric that measures the maximum difference between two cumulative distribution functions, commonly used to quantify convergence in distribution and in goodness-of-fit tests.

A. Kullback–Leibler divergence
Kullback–Leibler divergence is a fundamental information-theoretic measure that quantifies how one probability distribution differs from a reference distribution.
B. Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
C. Rényi divergence
Rényi divergence is a family of information-theoretic measures that generalize Kullback–Leibler divergence to quantify the dissimilarity between probability distributions, parameterized by an order α.
D. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
E. Rényi entropy
Rényi entropy is a generalized measure of information and uncertainty that extends Shannon entropy by introducing a tunable order parameter to emphasize different aspects of a probability distribution.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	metric on probability distributions ⓘ probability metric ⓘ statistical distance ⓘ
alsoKnownAs	Kolmogorov metric ⓘ linked to: Kolmogorov distance Kolmogorov–Smirnov distance ⓘ Kolmogorov–Smirnov metric ⓘ linked to: Kolmogorov distance
appliesTo	empirical distribution functions ⓘ theoretical distribution functions ⓘ
assumes	distributions defined on a common measurable space ⓘ
belongsTo	nonparametric goodness-of-fit methodology ⓘ theory of weak convergence of probability measures ⓘ
category	distance between probability distributions ⓘ
comparedWith	Lévy–Prokhorov metric ⓘ Wasserstein distance ⓘ total variation distance ⓘ
definedOn	cumulative distribution functions ⓘ probability distributions on the real line ⓘ
domainRestriction	distributions with cumulative distribution functions ⓘ
expressedAs	supremum over all real x of \|F(x) − G(x)\| ⓘ
invariantUnder	monotone transformations of the underlying variable that preserve order ⓘ
lessSensitiveTo	local deviations in tails compared to some other metrics ⓘ
mathematicalForm	d_K(F,G) = sup_x \|F(x) − G(x)\| ⓘ
measures	maximum difference between two cumulative distribution functions ⓘ
metricProperty	identity of indiscernibles ⓘ non-negativity ⓘ symmetry ⓘ triangle inequality ⓘ
namedAfter	Andrey Kolmogorov ⓘ linked to: Andrei Kolmogorov
relatedTo	Kolmogorov–Smirnov statistic ⓘ linked to: Kolmogorov distance Kolmogorov–Smirnov test ⓘ linked to: Kolmogorov distance supremum norm ⓘ uniform metric ⓘ
requires	right-continuous cumulative distribution functions with left limits ⓘ
sensitiveTo	global differences between distributions ⓘ
strongerThan	Lévy metric on the real line ⓘ linked to: Lévy–Prokhorov metric
topologyInduced	weak convergence of probability measures on the real line ⓘ
usedFor	goodness-of-fit testing ⓘ one-sample goodness-of-fit tests ⓘ quantifying convergence in distribution ⓘ two-sample comparison of distributions ⓘ
usedIn	Kolmogorov–Smirnov test ⓘ linked to: Kolmogorov distance nonparametric statistics ⓘ probability theory ⓘ statistical hypothesis testing ⓘ stochastic process convergence analysis ⓘ
usedToDefine	Kolmogorov–Smirnov statistic in empirical samples ⓘ

How these facts were elicited