Comonotonicity

A subset {{math|S}} of {{math|Rⁿ}} is called comonotonic{{harv|Kaas|Dhaene|Vyncke|Goovaerts|2002|loc=Definition 1}} (sometimes also nondecreasingSee {{harv|Nelsen|2006|loc=Definition 2.5.1}} for the case {{math|n {{=}} 2}}) if, for all {{math|(x₁, x₂, . . . , x_n)}} and {{math|(y₁, y₂, . . . , y_n)}} in {{math|S}} with {{math|x_i < y_i}} for some {{math|i ∈ {1, 2, . . . , n}}}, it follows that {{math|x_j ≤ y_j}} for all {{math|j ∈ {1, 2, . . . , n}}}.

This means that {{math|S}} is a totally ordered set.

Let {{math|μ}} be a probability measure on the {{math|n}}-dimensional Euclidean space {{math|Rⁿ}} and let {{math|F}} denote its multivariate cumulative distribution function, that is

:$F(x\_1,\backslash ldots,x\_n):=\backslash mu\backslash bigl(\backslash \{(y\_1,\backslash ldots,y\_n)\backslash in\{\backslash mathbb\; R\}^n\backslash mid\; y\_1\backslash le\; x\_1,\backslash ldots,y\_n\backslash le\; x\_n\backslash \}\backslash bigr),\backslash qquad\; (x\_1,\backslash ldots,x\_n)\backslash in\{\backslash mathbb\; R\}^n.$

Furthermore, let {{math|F₁, . . . , F_n}} denote the cumulative distribution functions of the {{math|n}} one-dimensional marginal distributions of {{math|μ}}, that means

:$F\_i(x):=\backslash mu\backslash bigl(\backslash \{(y\_1,\backslash ldots,y\_n)\backslash in\{\backslash mathbb\; R\}^n\backslash mid\; y\_i\backslash le\; x\backslash \}\backslash bigr),\backslash qquad\; x\backslash in\{\backslash mathbb\; R\}$

for every {{math|i ∈ {1, 2, . . . , n}}}. Then {{math|μ}} is called comonotonic, if

:$F(x\_1,\backslash ldots,x\_n)=\backslash min\_\{i\backslash in\backslash \{1,\backslash ldots,n\backslash \}\}F\_i(x\_i),\backslash qquad\; (x\_1,\backslash ldots,x\_n)\backslash in\{\backslash mathbb\; R\}^n.$

Note that the probability measure {{math|μ}} is comonotonic if and only if its support {{math|S}} is comonotonic according to the above definition.See {{harv|Nelsen|2006|loc=Theorem 2.5.4}} for the case {{math|n {{=}} 2}}

An {{math|Rⁿ}}-valued random vector {{math|X {{=}} (X₁, . . . , X_n)}} is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means

:$\backslash Pr(X\_1\backslash le\; x\_1,\backslash ldots,X\_n\backslash le\; x\_n)=\backslash min\_\{i\backslash in\backslash \{1,\backslash ldots,n\backslash \}\}\; \backslash Pr(X\_i\backslash le\; x\_i),\backslash qquad\; (x\_1,\backslash ldots,x\_n)\backslash in\{\backslash mathbb\; R\}^n.$

An {{math|Rⁿ}}-valued random vector {{math|X {{=}} (X₁, . . . , X_n)}} is comonotonic if and only if it can be represented as

:$(X\_1,\backslash ldots,X\_n)=\_\backslash text\{d\}(F\_\{X\_1\}^\{-1\}(U),\backslash ldots,F\_\{X\_n\}^\{-1\}(U)),\; \backslash ,$

where {{math|{{=}}_d}} stands for equality in distribution, on the right-hand side are the left-continuous generalized inverses{{harv|McNeil|Frey|Embrechts|2005|loc=Proposition A.3 (properties of the generalized inverse)}} of the cumulative distribution functions {{math|F_X1, . . . , F_Xn}}, and {{math|U}} is a uniformly distributed random variable on the unit interval. More generally, a random vector is comonotonic if and only if it agrees in distribution with a random vector where all components are non-decreasing functions (or all are non-increasing functions) of the same random variable.{{harv|McNeil|Frey|Embrechts|2005|loc=Proposition 5.16 and its proof}}

{{main|Fréchet–Hoeffding copula bounds}}

Let {{math|X {{=}} (X₁, . . . , X_n)}} be an {{math|Rⁿ}}-valued random vector. Then, for every {{math|i ∈ {1, 2, . . . , n}}},

:$\backslash Pr(X\_1\backslash le\; x\_1,\backslash ldots,X\_n\backslash le\; x\_n)\; \backslash le\; \backslash Pr(X\_i\backslash le\; x\_i),\backslash qquad\; (x\_1,\backslash ldots,x\_n)\backslash in\{\backslash mathbb\; R\}^n,$

hence

:$\backslash Pr(X\_1\backslash le\; x\_1,\backslash ldots,X\_n\backslash le\; x\_n)\backslash le\backslash min\_\{i\backslash in\backslash \{1,\backslash ldots,n\backslash \}\}\; \backslash Pr(X\_i\backslash le\; x\_i),\backslash qquad\; (x\_1,\backslash ldots,x\_n)\backslash in\{\backslash mathbb\; R\}^n,$

with equality everywhere if and only if {{math|(X₁, . . . , X_n)}} is comonotonic.

Let {{math|(X, Y)}} be a bivariate random vector such that the expected values of {{math|X}}, {{math|Y}} and the product {{math|XY}} exist. Let {{math|(X^*, Y^*)}} be a comonotonic bivariate random vector with the same one-dimensional marginal distributions as {{math|(X, Y)}}.{{math|(X^*, Y^*)}} always exists, take for example {{math|(F_X⁻¹(U), F_Y ⁻¹(U))}}, see section Properties above. Then it follows from Höffding's formula for the covariance{{harv|McNeil|Frey|Embrechts|2005|loc=Lemma 5.24}} and the upper Fréchet–Hoeffding bound that

:$\backslash text\{Cov\}(X,Y)\backslash le\backslash text\{Cov\}(X^*,Y^*)$

and, correspondingly,

:$\backslash operatorname\; E[XY]\backslash le\; \backslash operatorname\; E[X^*Y^*]$

with equality if and only if {{math|(X, Y)}} is comonotonic.{{harv|McNeil|Frey|Embrechts|2005|loc=Theorem 5.25(2)}}

Note that this result generalizes the rearrangement inequality and Chebyshev's sum inequality.

• Copula (probability theory)

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Category:Theory of probability distributions

Category:Independence (probability theory)

Category:Covariance and correlation