In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics, a copula is a multivariate
cumulative distribution function for which the
marginal probability distribution of each variable is
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
on the interval
, 1 Copulas are used to describe/model the
dependence (inter-correlation) between
random variables. Their name, introduced by applied mathematician
Abe Sklar
Abe Sklar (November 25, 1925 – October 30, 2020) was an American mathematician and a professor of applied mathematics at the Illinois Institute of Technology (IIT) and the inventor of copulas in probability theory.
Education and career
Sklar ...
in 1959, comes from the Latin for "link" or "tie", similar but unrelated to grammatical
copulas in
linguistics
Linguistics is the science, scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure ...
. Copulas have been used widely in
quantitative finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
to model and minimize tail risk
and
portfolio-optimization applications.
Sklar's theorem states that any multivariate
joint distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
can be written in terms of univariate
marginal distribution functions and a copula which describes the dependence structure between the variables.
Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulae separately. There are many parametric copula families available, which usually have parameters that control the strength of dependence. Some popular parametric copula models are outlined below.
Two-dimensional copulas are known in some other areas of mathematics under the name ''permutons'' and ''doubly-stochastic measures''.
Mathematical definition
Consider a random vector
. Suppose its marginals are continuous, i.e. the marginal
CDFs are
continuous functions. By applying the
probability integral transform In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to random ...
to each component, the random vector
:
has marginals that are
uniformly distributed on the interval
, 1
The copula of
is defined as the
joint cumulative distribution function of
:
:
The copula ''C'' contains all information on the dependence structure between the components of
whereas the marginal cumulative distribution functions
contain all information on the marginal distributions of
.
The reverse of these steps can be used to generate
pseudo-random samples from general classes of
multivariate probability distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
s. That is, given a procedure to generate a sample
from the copula function, the required sample can be constructed as
:
The inverses
are unproblematic
almost surely, since the
were assumed to be continuous. Furthermore, the above formula for the copula function can be rewritten as:
:
Definition
In
probabilistic terms,
.
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