General properties

Let $F$ be a continuous $d$-dimensional distribution function with marginal distributions $F_{1},\ldots,F_{d}$. Then:

There exists a unique copula function $C$ with $$\begin{equation} F(x_{1},\ldots,x_{d})=C(F_{1}(x_{1},\ldots,x_{d})) \end{equation}$$ for all $x_{1},\ldots,x_{d}\in \mathbb{R}^{d}$.

[Sklar’s theorem] Let $F$ be a continuous $d$ -dimensional distribution function with marginal distributions $F_{1},\ldots,F_{d}$ . Then:

There exists a unique copula function $C$ with $\begin{equation} F(x_{1},\ldots,x_{d})=C(F_{1}(x_{1},\ldots,x_{d})) \end{equation}$ for all $x_{1},\ldots,x_{d}\in \mathbb{R}^{d}$ .

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H-volume

Conditional probabilities

Definition h- and v-functions

$\begin{aligned} h(u_{1},u_{2}):&=C(u_{1}|u_{2})\\ &=\mathbb{P}(U_{1}\leq u_{1}|U_{2}=u_{2})\\ &=\frac{\partial C(u_{1},u_{2})}{\partial u_{2}}\\ &=\partial_{2}C(u_{1},u_{2}) \end{aligned}$ $\begin{aligned} v(u_{2},u_{1}):&=C(u_{2}|u_{1})\\ &=\mathbb{P}(U_{2}\leq u_{2}|U_{1}=u_{1})\\ &=\frac{\partial C(u_{1},u_{2})}{\partial u_{2}}\\ &=\partial_{1}C(u_{1},u_{2}) \end{aligned}$

Copula theory

Notes on copula theory

General properties

H-volume

Conditional probabilities

Definition h- and v-functions