Lévy metric - Alchetron, The Free Social Encyclopedia

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 F , G : R → [ 0 , 1 ] be two cumulative distribution functions. Define the Lévy distance between them to be

L ( F , G ) := inf { ε > 0 | F ( x − ε ) − ε ≤ G ( x ) ≤ F ( x + ε ) + ε f o r a l l x ∈ 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).

References

Lévy metric Wikipedia

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