Copulas

Overview

Theorem

A function {% G(u,v) : A \times B \rightarrow \mathbb{R} %} that is both grounded and 2-increasing is non-decreasing in {% u %} and {% v %}

This implies the following: For a sucopula {% C %} and for every {% (u,v) \in A \times B %}

Theorem

For a sub-copula {% C %}, {% C %} is uniformly continuous on {% A \times B %}.

Also for constants {% k_A %} and {% k_B %} the functions {% C(k_A) %} and {% C(k_B) %} are non-decreasing and uniformly continuous.

Theorem

The partial derivatives of a subcopula {% C %}, {% \frac{\partial C}{\partial u} %} and {% \frac{\partial C}{\partial v} %} exist almost everyhwere in the interior of {% A \times B %} and have values in the {% I = [0,1] %}.

Theorem

Sklar's Theorem

Given two cumulative distribution functions, {% F_1(x) %} and {% F_2(x) %}

• For a subcopula whose domain includes the range of {% F_1(x) %} and {% F_2(y) %}
  {% F(x,y) = C(F_1(x), F_2(y)) %}
  is a joint distribution function where {% F_1(x) %} and {% F_2(y) %} are the marginal distributions.
• Likewise given marginal distributions {% F_1(x) %} and {% F_2(y) %} there exists a unique subcopula such that
  {% F(x,y) = C(F_1(x), F_2(y)) %}