[Math] the motivation of Levy-Prokhorov metric

measure-theorymetric-spaces

From Wikipedia

Let $(M, d)$ be a metric space with its Borel sigma algebra
$\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all
probability measures on the measurable space $(M, \mathcal{B} (M))$.

For a subset $A \subseteq M$, define the $ε$-neighborhood of $A$ by $$
A^{\varepsilon} := \{ p \in M ~|~ \exists q \in A, \ d(p, q) < \varepsilon \} = \bigcup_{p \in A} B_{\varepsilon} (p). $$ where
$B_{\varepsilon} (p)$ is the open ball of radius $\varepsilon$
centered at $p$.

The Lévy–Prokhorov metric $\pi : \mathcal{P} (M)^{2} \to [0, +
\infty)$ is defined by setting the distance between two probability
measures $\mu$ and $\nu$ to be $$
\pi (\mu, \nu) := \inf \left\{ \varepsilon > 0 ~|~ \mu(A) \leq \nu (A^{\varepsilon}) + \varepsilon \ \text{and} \ \nu (A) \leq \mu
(A^{\varepsilon}) + \varepsilon \ \text{for all} \ A \in
\mathcal{B}(M) \right\}. $$

  1. I wonder what the purpose, motivation and intuition of the L-P metric are?
  2. Is the following alternative a reasonable metric or some generalized metric between
    measures $$ \sup_{A \in \mathcal{B}(M)} |\mu(A) – \nu(A)|? $$ If
    yes, is this one more simple and easy to understand and therefore maybe more useful than L-P
    metric?

  3. A related metric between distribution functions is the Levy metric:

    Let $F, G : \mathbb{R} \to [0, + \infty)$ be two cumulative
    distribution functions. Define the Lévy distance between them to be $$
    L(F, G) := \inf \{ \varepsilon > 0 | F(x – \varepsilon) – \varepsilon \leq G(x) \leq F(x + \varepsilon) + \varepsilon
    \mathrm{\,for\,all\,} x \in \mathbb{R} \}. $$

    I wonder how to picture this intuition part:

    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 $L(F, G)$.

Thanks and regards!

Best Answer

Most of what occurs to me has already been said, but you may find the following picture useful.

If $d_C$ is the Chebyshev metric on $R^2$, i.e. with points $\mathbf{p} = (x_1,y_1)$ and $\mathbf{q} = (x_2,y_2)$ in $R^2$,

$d_C(\mathbf{p,q}) := |x_1-x_2| \vee |y_1-y_2|$,

and $h_C$ is the Hausdorff metric on closed subsets of $R^2$ induced by $d_C$, i.e. with $A$ and $B$ being closed subsets of $R^2$,

$h_C(A,B):= \sup_{\mathbf{p} \in A} d_C(\mathbf{p},B) \vee \sup_{\mathbf{q} \in B} d_C(\mathbf{q},A)$,

where as usual $d_C(\mathbf{p},B) = \inf_{\mathbf{r} \in B} d_C(\mathbf{p,r})$ etc,

then the Levy metric between two distribution functions $F$ and $G$ is simply the Hausdorff distance $d_C$ between the closures of the completed graphs of $F$ and $G$.

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