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
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ever ...
for which the
marginal probability
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variab ...
distribution of each variable is
uniform 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. Copulas have been used widely in
quantitative finance to model and minimize tail risk
and
portfolio-optimization applications.
Sklar's theorem states that any multivariate
joint distribution can be written in terms of univariate
marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variab ...
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 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
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process.
Background
The generation of random numbers has many uses, such as for rando ...
samples from general classes of
multivariate probability distributions. 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
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
, since the
were assumed to be continuous. Furthermore, the above formula for the copula function can be rewritten as:
:
Definition
In
probabilistic terms,
.
The_following_tables_highlight_the_most_prominent_bivariate_Archimedean_copulas,_with_their_corresponding_generator._Not_all_of_them_are_completely_monotone_function.html" ;"title="convex_function.html" "title="d-monotone_function.html" ;"title=",\infty) is a continuous, strictly decreasing and convex function such that