Английская Википедия:Comonotonicity

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In probability theory, comonotonicity mainly refers to the perfect positive dependence between the components of a random vector, essentially saying that they can be represented as increasing functions of a single random variable. In two dimensions it is also possible to consider perfect negative dependence, which is called countermonotonicity.

Comonotonicity is also related to the comonotonic additivity of the Choquet integral.

The concept of comonotonicity has applications in financial risk management and actuarial science, see e.g. Шаблон:Harvtxt and Шаблон:Harvtxt. In particular, the sum of the components Шаблон:Math is the riskiest if the joint probability distribution of the random vector Шаблон:Math is comonotonic. Furthermore, the Шаблон:Math-quantile of the sum equals of the sum of the Шаблон:Math-quantiles of its components, hence comonotonic random variables are quantile-additive. In practical risk management terms it means that there is minimal (or eventually no) variance reduction from diversification.

For extensions of comonotonicity, see Шаблон:Harvtxt and Шаблон:Harvtxt.

Definitions

Comonotonicity of subsets of Шаблон:Math

A subset Шаблон:Math of Шаблон:Math is called comonotonic (sometimes also nondecreasing) if, for all Шаблон:Math and Шаблон:Math in Шаблон:Math with Шаблон:Math for some Шаблон:Math}, it follows that Шаблон:Math for all Шаблон:Math}.

This means that Шаблон:Math is a totally ordered set.

Comonotonicity of probability measures on Шаблон:Math

Let Шаблон:Math be a probability measure on the Шаблон:Math-dimensional Euclidean space Шаблон:Math and let Шаблон:Math denote its multivariate cumulative distribution function, that is

<math>F(x_1,\ldots,x_n):=\mu\bigl(\{(y_1,\ldots,y_n)\in{\mathbb R}^n\mid y_1\le x_1,\ldots,y_n\le x_n\}\bigr),\qquad (x_1,\ldots,x_n)\in{\mathbb R}^n.</math>

Furthermore, let Шаблон:Math denote the cumulative distribution functions of the Шаблон:Math one-dimensional marginal distributions of Шаблон:Math, that means

<math>F_i(x):=\mu\bigl(\{(y_1,\ldots,y_n)\in{\mathbb R}^n\mid y_i\le x\}\bigr),\qquad x\in{\mathbb R}</math>

for every Шаблон:Math}. Then Шаблон:Math is called comonotonic, if

<math>F(x_1,\ldots,x_n)=\min_{i\in\{1,\ldots,n\}}F_i(x_i),\qquad (x_1,\ldots,x_n)\in{\mathbb R}^n.</math>

Note that the probability measure Шаблон:Math is comonotonic if and only if its support Шаблон:Math is comonotonic according to the above definition.

Comonotonicity of Шаблон:Math-valued random vectors

An Шаблон:Math-valued random vector Шаблон:Math is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means

<math>\Pr(X_1\le x_1,\ldots,X_n\le x_n)=\min_{i\in\{1,\ldots,n\}} \Pr(X_i\le x_i),\qquad (x_1,\ldots,x_n)\in{\mathbb R}^n.</math>

Properties

An Шаблон:Math-valued random vector Шаблон:Math is comonotonic if and only if it can be represented as

<math>(X_1,\ldots,X_n)=_\text{d}(F_{X_1}^{-1}(U),\ldots,F_{X_n}^{-1}(U)), \, </math>

where Шаблон:Math stands for equality in distribution, on the right-hand side are the left-continuous generalized inverses of the cumulative distribution functions Шаблон:Math, and Шаблон:Math 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.

Upper bounds

Upper Fréchet–Hoeffding bound for cumulative distribution functions

Let Шаблон:Math be an Шаблон:Math-valued random vector. Then, for every Шаблон:Math},

<math>\Pr(X_1\le x_1,\ldots,X_n\le x_n) \le \Pr(X_i\le x_i),\qquad (x_1,\ldots,x_n)\in{\mathbb R}^n,</math>

hence

<math>\Pr(X_1\le x_1,\ldots,X_n\le x_n)\le\min_{i\in\{1,\ldots,n\}} \Pr(X_i\le x_i),\qquad (x_1,\ldots,x_n)\in{\mathbb R}^n,</math>

with equality everywhere if and only if Шаблон:Math is comonotonic.

Upper bound for the covariance

Let Шаблон:Math be a bivariate random vector such that the expected values of Шаблон:Math, Шаблон:Math and the product Шаблон:Math exist. Let Шаблон:Math be a comonotonic bivariate random vector with the same one-dimensional marginal distributions as Шаблон:Math. Then it follows from Höffding's formula for the covariance and the upper Fréchet–Hoeffding bound that

<math>\text{Cov}(X,Y)\le\text{Cov}(X^*,Y^*)</math>

and, correspondingly,

<math> \operatorname E[XY]\le \operatorname E[X^*Y^*]</math>

with equality if and only if Шаблон:Math is comonotonic.

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

See also

• Copula (probability theory)

References

• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation