---
tag: BJ33Y5H49M
kind: definition
name: Copula
msc: [62H05, "60E05"]
defines: CAMJRBZANK
uses: [0H8H7ZTHW2, 0MXT6PRZZ6, 0AREFQ9Z65, 0DR81R1F2B]
sources:
  - url: https://en.wikipedia.org/wiki/Copula_(statistics)
---

## Definition

Let $d \ge 2$.
A *$d$-dimensional copula* is a function $C \colon [0, 1]^d \to [0, 1]$ such that there is a random vector $U = (U_1, \ldots, U_d)$ with:

1. each $U_i$ [[uniformly distributed|0AREFQ9Z65]] on $[0, 1]$, and
2. $C(u_1, \ldots, u_d) = P(U_1 \le u_1, \ldots, U_d \le u_d)$ for all $(u_1, \ldots, u_d) \in [0, 1]^d$.

So a copula is the restriction to $[0, 1]^d$ of the joint [[cumulative distribution function|0H8H7ZTHW2]] of a random vector with uniform [[marginals|0MXT6PRZZ6]].

## Remarks

Example: if $U_1, \ldots, U_d$ are independent and uniform on $[0, 1]$, then $C(u) = u_1 u_2 \cdots u_d$, the independence copula.
Example: if $U_1 = U_2 = U$ with $U$ uniform, then $C(u_1, u_2) = P(U \le \min(u_1, u_2)) = \min(u_1, u_2)$.
Non-example: $H(u_1, u_2) = u_1^2 u_2^2$ on $[0, 1]^2$ is the joint distribution function of two independent variables with distribution function $t \mapsto t^2$; its marginals are not uniform, since $H(u_1, 1) = u_1^2$, so it is not a copula.
Equivalently, $C$ is a copula when it is grounded ($C(u) = 0$ if some $u_i = 0$), satisfies $C(1, \ldots, 1, u_i, 1, \ldots, 1) = u_i$, and is $d$-increasing (it gives non-negative mass to every box).
Every copula satisfies the Fréchet–Hoeffding bounds $\max(u_1 + \cdots + u_d - d + 1, 0) \le C(u) \le \min(u_1, \ldots, u_d)$.