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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