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A function that links a multi-dimensional [[Probability distribution|probability distribution]] function to its one-dimensional margins. Such functions first made their appearance in the work of M. Fréchet, W. Höffding, R. Féron, and G. Dall'Aglio. However, their explicit definition and the recognition that they are important in their own right is due to A. Sklar. Presently (1996), the best sources for information are [[#References|[a1]]] and [[#References|[a2]]].

A (two-dimensional) copula is a function  $  C $
from the unit square  $  [ 0,1 ] \times [ 0,1 ] $
onto the unit interval  $  [ 0,1 ] $
such that:

1)  $  C ( a,0 ) = C ( 0,a ) = 0 $
and  $  C ( a,1 ) = C ( 1,a ) = a $
for any  $  a \in [ 0,1 ] $;

2)  $  C ( a _ {2} ,b _ {2} ) - C ( a _ {1} ,b _ {2} ) - C ( a _ {2} ,b _ {1} ) + C ( a _ {1} ,b _ {1} ) \geq  0 $
whenever  $  a _ {1} \leq  a _ {2} $
and  $  b _ {1} \leq  b _ {2} $.

If  $  C $
is a copula, then  $  C $
is non-decreasing in each place and continuous, and hence a continuous bivariate [[Distribution function|distribution function]] on the unit square, with uniform margins. Furthermore, setting  $  W ( a,b ) =  \max  ( a + b - 1,0 ) $, 
one has  $  W ( a,b ) \leq  C ( a,b ) \leq   \min  ( a,b ) $
for all  $  a,b $
in  $  [ 0,1 ] $. 
The functions  $  W $
and  $   \min  $
are copulas, as is the function  $  \pi $
given by  $  \pi ( a,b ) = ab $.

The central Sklar theorem states that if  $  H $
is a two-dimensional distribution function with one-dimensional marginal distribution functions  $  F $
and  $  G $, 
then there exists a copula  $  C $
such that for all  $  x,y \in \mathbf R $,

$$ 
H ( x,y ) = C ( F ( x ) ,G ( y ) ) .
$$

If  $  F $
and  $  G $
are continuous, then  $  C $
is unique; otherwise  $  C $
is uniquely determined on  $  ( { \mathop{\rm range} } F ) \times ( { \mathop{\rm range} } G ) $. 
It follows that if  $  X $
and  $  Y $
are real random variables (cf. [[Random variable|Random variable]]) with distribution functions  $  F _ {X} $
and  $  F _ {Y} $
and joint distribution function  $  H _ {XY }  $, 
then there is a copula  $  C _ {XY }  $
such that  $  H _ {XY }  ( x,y ) = C _ {XY }  ( F _ {X} ( x ) ,F _ {Y} ( y ) ) $. 
The random variables  $  X $, 
$  Y $
are independent if and only if it is possible to take  $  C = \Pi $.

Sklar's theorem shows that much of the study of joint distribution functions can be reduced to the study of copulas. Furthermore, under a.s. strictly increasing transformations of  $  X $
and  $  Y $, 
the copula  $  C _ {XY }  $
is invariant, while the margins may be changed at will. Thus (for random variables with continuous distribution functions) the study of rank statistics (insofar as it is the study of properties invariant under increasing transformations, cf. [[Rank statistic|Rank statistic]]) may be characterized as the study of copulas and copula-invariant properties.

For random variables with continuous distribution functions, the extreme copulas  $   \min  $
and  $  W $
are attained precisely when  $  X $
is a.s. an increasing (respectively, decreasing) function of  $  Y $. 
Hence, copulas can be used to construct non-parametric measures of dependence. One such is the quantity

$$ 
\sigma ( x,y ) = 12 \int\limits _ { 0 } ^ { 1 }  \int\limits _ { 0 } ^ { 1 }  {\left | {C _ {XY }  ( u,v ) - uv } \right | }  {d u }  {dv } .
$$

Furthermore, in terms of copulas, the two best known non-parametric measures of dependence, namely Spearman's measure of dependence  $  \rho $
and Kendall's measure of dependence  $  \tau $, 
are given by

$$ 
\rho ( X,Y ) = 12 \int\limits _ { 0 } ^ { 1 }  \int\limits _ { 0 } ^ { 1 }  {( C _ {XY }  ( u,v ) - uv ) }  {du }  {dv }
$$

and

$$ 
\tau ( X,Y ) = 4 \int\limits _ { 0 } ^ { 1 }  \int\limits _ { 0 } ^ { 1 }  {C _ {XY }  ( u,v ) }  {d C _ {XY }  ( u,v ) } - 1;
$$

and the fact that  $  X $
and  $  Y $
are positively quadrant dependent is succinctly expressed by the condition  $  C _ {XY }  \geq  \Pi $.

Several classes of copulas merit special attention. First, there are the Archimedean copulas, which admit the representation  $  C ( u,v ) = h ^ {- 1 } ( h ( u ) + h ( v ) ) $
with  $  h $
a continuous decreasing convex function from  $  [ 0,1 ] $
into  $  [ 0, \infty ] $
satisfying  $  h ( 1 ) = 0 $. 
These may be used to generate various (generally, one- or two-parameter) families of bivariate distribution functions, and, as a consequence, play an important role in modelling non-normal dependence and testing for such dependence [[#References|[a4]]].

Next, there are the shuffles of  $   \min  $. 
These are obtained by redistributing the mass distribution of  $   \min  $(
which is uniformly distributed on the main diagonal of the unit square) in such a way that the resultant mass distribution remains singular. These shuffles are dense in the space of all copulas. Nevertheless, it is still true that if  $  X $
and  $  Y $
are random variables whose copula is a shuffle of  $   \min  $, 
then there is an invertible function  $  g $
such that  $  Y = g ( X ) $. 
This yields the striking fact that for any pair of independent random variables  $  X,Y $
there is a pair of random variables  $  U,V $
having the same individual distribution functions as  $  X,Y $
and having a copula arbitrarily close to  $  \Pi $, 
but such that each is completely determined by the other. (See the Mikusińksi–Sherwood–Taylor paper in [[#References|[a1]]].)

Lastly, a copula determines a doubly-stochastic measure on the unit square. Such measures have been of interest for a long time and considerable effort has been devoted to finding extreme points of this convex set. Here, an approach using copulas has led to several new classes of such extreme points, the hairpins and generalized hairpins, as well as to further insight into the general problem. (See the Mikusińksi–Sherwood–Taylor paper in [[#References|[a1]]].)

Let  $  * $
be the binary operation defined on the set of two-dimensional copulas by

$$ 
( A * B ) ( u,v ) = \int\limits _ { 0 } ^ { 1 }  {A _ {,2 }  ( u,t ) B _ {,1 }  ( t,v ) }  {dt } ,
$$

where  $  A _ {,2 }  $
denotes the partial derivative of  $  A $
with respect to its second argument and  $  B _ {,1 }  $
the partial derivative of  $  B $
with respect to its first argument (these partial derivatives exists almost everywhere). Then  $  A * B $
is a copula, and the set of copulas is a [[Semi-group|semi-group]] under the operation  $  * $. 
The salient fact concerning this operation is the following: If  $  \{ {X _ {t} } : {t \in T } \} $
is a real [[Stochastic process|stochastic process]] with parameter set  $  T $
and if  $  C _ {st }  $
is the copula of  $  X _ {s} $
and  $  X _ {t} $, 
then the transition probabilities of the process satisfy the [[Kolmogorov–Chapman equation|Kolmogorov–Chapman equation]] if and only if  $  C _ {st }  = C _ {su }  * C _ {ut }  $
for all  $  s,u,t \in T $
such that  $  s < u < t $. 
This result is the key to a new approach to the theory of Markov processes (cf. [[Markov process|Markov process]]) and to a new way of constructing them. It also leads to an interesting area of functional analysis: the study of Markov algebras, [[#References|[a3]]].

Finally, the concept of a copula can be extended to  $  n $
dimensions. An  $  n $-
copula may be viewed as an  $  n $-
dimensional distribution function whose support is in the unit  $  n $-
cube and whose one-dimensional margins are uniform. If  $  H $
is an  $  n $-
dimensional distribution function with one-dimensional margins  $  F _ {1} \dots F _ {n} $, 
then there is an  $  n $-
copula  $  C $
such that

$$ 
H ( x _ {1} \dots x _ {n} ) = C ( F _ {1} ( x _ {1} ) \dots F _ {n} ( x _ {n} ) )
$$

for all  $  ( x _ {1} \dots x _ {n} ) \in \mathbf R  ^ {n} $. 
Moreover, for any  $  n $-
copula:

$$ 
 \max  ( x _ {1} + \dots + x _ {n} - n + 1,0 ) \leq 
$$

$$ 
\leq  
C ( x _ {1} \dots x _ {n} ) \leq   \min  ( x _ {1} \dots x _ {n} ) ;
$$

however, while the upper function is an  $  n $-
copula for any  $  n $, 
the lower function is not an  $  n $-
copula for any  $  n > 2 $.

A basic problem in the theory of copulas is that of compatibility, i.e., to determine which sets of copulas (of possible different dimensions) can appear as margins of a single higher-dimensional copula.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  "Advances in probability distributions with given marginals: beyond the copulas"  G. Dall'Aglio (ed.)  S. Kotz (ed.)  G. Salinetti (ed.) , Kluwer Acad. Publ.  (1991)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  "Distributions with fixed marginals and related topics"  L. Rüschendorf (ed.)  B. Schweizer (ed.)  M.D. Taylor (ed.) , ''Lecture Notes Monograph Ser.'' , '''28''' , Inst. Math. Stat.  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.F. Darsow,   B. Nguyen,   E.T. Olsen,   "Copulas and Markov processes"  ''Ill. J. Math.'' , '''36'''  (1992)  pp. 600–642</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Genest,   L.-P. Rivest,   "Statistical inference procedures for bivariate Archimedean copulas"  ''J. Amer. Statist. Assoc.'' , '''88'''  (1993)  pp. 1034–1043</TD></TR></table>