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Copula in Statistics

The cumulative distribution function of a random variable $X=(X_1,\dots,X_N)$ is: $F(x_1,\dots,x_N):=\mathbb P(X_1\leq x_1,\dots,X_N\leq x_N)$.

The copula of $X$ is $C:[0,1]^N\to[0,1]$ such that: $F(x_1,\dots,x_N)=C(F_1(x_1),\dots, F_N(X_N))$ where $F$, resp. $F_i$, is the distribution function of $X$, resp. $X_i$. It is unique if each variable $X_i$ is continuous (atomless law).

Theorem (Sklar). A function $C:[0,1]^N\to[0,1]$ is the copula of some random variable with values in $\mathbb R^N$ if and only if the following properties are satisfied for all $i$: (i) $C(x_1,\dots,x_{i-1},0,x_{i+1},\dots,u_N)=0$; (ii) $C(1,\dots,1,x_i,1,\dots,1)=u_i$; (iii) $\sum_{t\in\{1,2\}^N} (-1)^{\sum_i t_j} C(x_1^{t_1},\dots,x_N^{t_N})\geq 0$ where $x_i^1\leq x_i^2$ for all.

Remark. A function $C:[0,1]^N\to[0,1]$ is a copula iff it is the (restriction to $[0,1]^N$ of the) repartition function of a random vector $(U_1,\dots,U_N)$ where each $U_i$ is uniform over $[0,1]$. $(X_1,\dots,X_N)$ is independent iff it admits the copula $C(u_1,\dots,u_N)=u_1\dots u_N$.

For an uniform random variable $U$, the vector $(U,\dots,U)$ admits the copula $C(u_1,\dots,u_N)=\min(u_1,\dots,u_N)$.

Fréchet bounds: Any copula $C$ satisfies: $\left(\sum_i u_i-n+1\right)^+ \leq C(u_1,\dots,u_N) \leq \min(u_1,\dots,u_N)$

The left hand side is itself a copula only for $N=2$.

The restriction of a copula to fewer variables is again a copula.There is no known general way to extend a copula to more variables.

See wikipedia.