Fréchet principle

The Fréchet principle , named after the French mathematician Maurice René Fréchet (1878–1973), is a general recipe for defining the expected value and variance for random variables in a suitable manner . The principle goes back to a work by Fréchet from 1948.

Definitions

Let be a metric space , where a metric is on the basic set . Let further be a random variable with values ​​in . Then the Fréchet expectation value is defined as the value that minimizes the expected square metric distance between and the elements , i.e. H. ${\displaystyle (M,d)}$ ${\displaystyle d}$ ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle EX}$ ${\displaystyle X}$ ${\displaystyle a\in M}$

${\displaystyle Ed^{2}(X,EX)=\min _{a\in M}Ed^{2}(X,a)}$.

This minimum achievable expected square distance defines the variance, ie

${\displaystyle VarX:=Ed^{2}(X,EX)}$.

Examples

• For the Euclidean metric and a classical random variable fulfill and the Fréchet principle, d. H. it applies ${\displaystyle M=\mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle EX}$ ${\displaystyle VarX}$

${\displaystyle VarX=E(X-EX)^{2}=\min _{a\in \mathbb {R} }E(X-a)^{2}}$.

• It is a bit more complicated in the case of random sets or random fuzzy sets . The Hausdorff metric is often used there as a metric , but the i. General did not use Aumann expectation as Fréchet expectation with respect to . This is not suitable for meaningfully defining a variance for random (fuzzy) sets. At least for random convex (fuzzy) sets, however, the Fréchet expectation value with respect to a metric defined by the carrier functions of convex sets. This then naturally gives the variance of a random convex (fuzzy) set. ${\displaystyle h}$ ${\displaystyle E_{A}}$ ${\displaystyle h}$ ${\displaystyle h}$ ${\displaystyle E_{A}}$

Individual evidence

1. ↑ Fréchet, M. (1948). Les éléments aléatoires de natures quelconque dans un éspace distancié . Ann.Inst.Poincaré 10, 215-310
2. ↑ Koerner, R. (1997). On the variance of fuzzy random variables . Fuzzy Sets and Systems 92, 83-93