April 25, 2024
Copulas let you glue together marginal distributions to get joint distributions. If the random variable X has cumulative distribution function F(x) = P(X\le x) then F^{-1}(X) is uniformly distributed on the unit interval [0,1]. Given random variables X_i, 1\le i\le n, with cdfs F_i and a measure C on the unit cube [0,1]^n the we can define a joint distribution P(X_1\le x_1, \ldots, X_n\le x_n) = C(F_1^{-1}(x_1), \ldots F_n^{-1}(x_n)) where C(u_1, \ldots, u_n) is the measure of [0,u_1]\times\cdots\times [0,u_n].
In the case of two random variables the copula of X and Y is the joint distribution of F^{-1}(X) and G^{-1}(Y) where F and G are the cumulative distributions of X and Y respectively: C(u,v) = C^{X,Y}(u,v) = P(F^{-1}(X) \le u, G^{-1}(Y) \le v).
Exercise: Show C(u,v) = H(F(u),G(v)) where and H is the joint distribution of X and Y and F and G are the marginal distributions of X, and Y.
Exercise: Show H(x,y) = C(F^{-1}(x), G^{-1}(y)).
This shows how to use the copula and marginal distributions to recover the joint distribution.
An equivalent definition is a copula is that it is a probability measure on [0,1]^2 with uniform marginals.
Exercise: Prove this.
If U and V are independent, uniformly distributed random variables on the unit interval then C(u,v) = uv.
If V=U then their joint distribution is C(u,v) = P(U\le u, V\le v) = P(U\le u, U\le v) = P(U\le \min\{u, v\}) = \min\{u,v\} = M(u,v).
If V=1-U then their joint distribution is C(u,v) = P(U\le u, V\le v) = P(U\le u, 1-U\le v) = P(1-v\le U\le u) = \max\{u - (1 -v), 0\} = \max\{u + v - 1, 0\} = W(u,v)
Exercise: (Fréchet-Hoeffding) For every copula, C, W \le C \le M.
Hint: For the upper bound use H(x,y) \le F(x) and H(x,y) \le G(y). For the lower bound note 0\le C(u_1,v_1) - C(u_1, v_2) - C(u_2, v_1) + C(u_2, v_2) for u_1 \ge u_2 and v_1 \ge v_2.