Lévy metric

Lévy metric

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 Pierre Lévy.

Definition

Let F, G : mathbb{R} o [0, + infty) be two cumulative distribution functions. Define the Lévy distance between them to be: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} }.

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).

ee also

* Lévy-Prokhorov metric
* Wasserstein metric

References

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