# 耦合 (概率)

## 定义

${\displaystyle C\left(\mathbf {u} \right)=0}$${\displaystyle \mathbf {u} \in [0,1]^{n}}$有至少一个分量为${\displaystyle 0;}$
${\displaystyle C\left(\mathbf {u} \right)=u_{i}}$${\displaystyle \mathbf {u} \in [0,1]^{n}}$所有分量为${\displaystyle 1}$除了第i个分量等于${\displaystyle u_{i};}$
${\displaystyle C\left(\mathbf {u} \right)}$是n维递增的，也即，有每个hyperrectangle ${\displaystyle B=\times _{i=1}^{n}[x_{i},y_{i}]\subseteq [0,1]^{n};}$
${\displaystyle V_{C}\left(B\right):=\sum _{\mathbf {z} \in \times _{i=1}^{n}\{x_{i},y_{i}\}}(-1)^{N(\mathbf {z} )}C(\mathbf {z} )\geq 0;}$

## Sklar定理

${\displaystyle H(x,y)=C(F(x),G(y))\,}$

（此处已知分布C和它的累积分布函数）。此外，如果边缘分布Fx） 和Gy）连续，那么关联结构函数C是唯一的。否则，关联结构C在边缘分布的值域上是唯一确定的。

## 弗雷歇–霍夫丁（Fréchet–Hoeffding）关联结构边界

Graphs of the Fréchet–Hoeffding copula limits and of the independence copula (in the middle).

${\displaystyle W(u,v)=\max(0,u+v-1).\,}$

n-元关联结构，下边界为

${\displaystyle W(u_{1},\ldots ,u_{n}):=\max \left\{1-n+\sum \limits _{i=1}^{n}{u_{i}},0\right\}\leq C(u_{1},\ldots ,u_{n}).}$

${\displaystyle M(u,v)=\min(u,v).\,}$

n-元关联结构，上边界为

${\displaystyle C(u_{1},\ldots ,u_{n})\leq \min _{j\in \{1,\ldots ,n\}}u_{j}=:M(u_{1},\ldots ,u_{n}).}$

${\displaystyle W(u,v)\leq C(u,v)\leq M(u,v).}$

${\displaystyle W(u_{1},\ldots ,u_{n})\leq C(u_{1},\ldots ,u_{n})\leq M(u_{1},\ldots ,u_{n}).}$

## 关联结构种类

### 正态关联结构

Cumulative distribution and probability density functions of Gaussian copula with ρ = 0.4

${\displaystyle C_{\rho }(u,v)=\Phi _{\rho }\left(\Phi ^{-1}(u),\Phi ^{-1}(v)\right)}$

C微分得出关联结构的密度函数：

${\displaystyle c_{\rho }(u,v)={\frac {\varphi _{X,Y,\rho }(\Phi ^{-1}(u),\Phi ^{-1}(v))}{\varphi (\Phi ^{-1}(u))\varphi (\Phi ^{-1}(v))}}}$

${\displaystyle \varphi _{X,Y,\rho }(x,y)={\frac {1}{2\pi {\sqrt {1-\rho ^{2}}}}}\exp \left(-{\frac {1}{2(1-\rho ^{2})}}\left[{x^{2}+y^{2}}-2\rho xy\right]\right)}$

## 参考资料

1. ^ Sklar, A. Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris. 1959, 8: 229–231.