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In [[mathematics]], the '''Lévy metric''' is a [[metric (mathematics)|metric]] on the space of [[cumulative distribution function]]s of one-dimensional [[random variable]]s. It is a special case of the [[Lévy–Prokhorov metric]], and is named after the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]]. | |||
==Definition== | |||
Let <math>F, G : \mathbb{R} \to [0, 1]</math> be two cumulative distribution functions. Define the '''Lévy distance''' between them to be | |||
:<math>L(F, G) := \inf \{ \varepsilon > 0 | F(x - \varepsilon) - \varepsilon \leq G(x) \leq F(x + \varepsilon) + \varepsilon \mathrm{\,for\,all\,} x \in \mathbb{R} \}.</math> | |||
Intuitively, if between the graphs of ''F'' and ''G'' one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to ''L''(''F'', ''G''). | |||
==See also== | |||
* [[Càdlàg]] | |||
* [[Lévy–Prokhorov metric]] | |||
* [[Wasserstein metric]] | |||
==References== | |||
* {{springer|author=V.M. Zolotarev|id=l/l058310|title=Lévy metric}} | |||
{{DEFAULTSORT:Levy metric}} | |||
[[Category:Measure theory]] | |||
[[Category:Metric geometry]] | |||
[[Category:Probability theory]] | |||
Revision as of 12:55, 20 January 2014
In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy.
Definition
Let be two cumulative distribution functions. Define the Lévy distance between them to be
![F,G:{\mathbb {R}}\to [0,1]](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee4a677314b81bb5bd48ea0387a94259476a172)
Intuitively, if between the graphs of F and G one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to L(F, G).
See also
References
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