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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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104101.png" /> from the unit square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104102.png" /> onto the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104103.png" /> such that:
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104105.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104106.png" />;
+
1) $  C ( a,0 ) = C ( 0,a ) = 0 $
 +
and $  C ( a,1 ) = C ( 1,a ) = a $
 +
for any $  a \in [ 0,1 ] $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104107.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c1104109.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041010.png" /> is a copula, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041012.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041015.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041017.png" /> are copulas, as is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041018.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041019.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041020.png" /> is a two-dimensional distribution function with one-dimensional marginal distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041022.png" />, then there exists a copula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041023.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041024.png" />,
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041025.png" /></td> </tr></table>
+
$$
 +
H ( x,y ) = C ( F ( x ) ,G ( y ) ) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041027.png" /> are continuous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041028.png" /> is unique; otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041029.png" /> is uniquely determined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041030.png" />. It follows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041032.png" /> are real random variables (cf. [[Random variable|Random variable]]) with distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041034.png" /> and joint distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041035.png" />, then there is a copula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041036.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041037.png" />. The random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041039.png" /> are independent if and only if it is possible to take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041040.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041042.png" />, the copula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041043.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041045.png" /> are attained precisely when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041046.png" /> is a.s. an increasing (respectively, decreasing) function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041047.png" />. Hence, copulas can be used to construct non-parametric measures of dependence. One such is the quantity
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041048.png" /></td> </tr></table>
+
$$
 +
\sigma ( x,y ) = 12 \int\limits _ { 0 } ^ { 1 }  \int\limits _ { 0 } ^ { 1 }  {\left | {C _ {XY }  ( u,v ) - uv } \right | }  {d u }  {dv } .
 +
$$
  
Furthermore, in terms of copulas, the two best known non-parametric measures of dependence, namely Spearman's measure of dependence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041049.png" /> and Kendall's measure of dependence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041050.png" />, are given by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041051.png" /></td> </tr></table>
+
$$
 +
\rho ( X,Y ) = 12 \int\limits _ { 0 } ^ { 1 }  \int\limits _ { 0 } ^ { 1 }  {( C _ {XY }  ( u,v ) - uv ) }  {du }  {dv }
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041052.png" /></td> </tr></table>
+
$$
 +
\tau ( X,Y ) = 4 \int\limits _ { 0 } ^ { 1 }  \int\limits _ { 0 } ^ { 1 }  {C _ {XY }  ( u,v ) }  {d C _ {XY }  ( u,v ) } - 1;
 +
$$
  
and the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041054.png" /> are positively quadrant dependent is succinctly expressed by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041055.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041056.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041057.png" /> a continuous decreasing convex function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041058.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041059.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041060.png" />. 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]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041062.png" />. These are obtained by redistributing the mass distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041063.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041065.png" /> are random variables whose copula is a shuffle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041066.png" />, then there is an invertible function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041067.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041068.png" />. This yields the striking fact that for any pair of independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041069.png" /> there is a pair of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041070.png" /> having the same individual distribution functions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041071.png" /> and having a copula arbitrarily close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041072.png" />, but such that each is completely determined by the other. (See the Mikusińksi–Sherwood–Taylor paper in [[#References|[a1]]].)
+
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]]].)
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041073.png" /> be the binary operation defined on the set of two-dimensional copulas by
+
Let $  * $
 +
be the binary operation defined on the set of two-dimensional copulas by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041074.png" /></td> </tr></table>
+
$$
 +
( A * B ) ( u,v ) = \int\limits _ { 0 } ^ { 1 }  {A _ {,2 }  ( u,t ) B _ {,1 }  ( t,v ) }  {dt } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041075.png" /> denotes the partial derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041076.png" /> with respect to its second argument and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041077.png" /> the partial derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041078.png" /> with respect to its first argument (these partial derivatives exists almost everywhere). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041079.png" /> is a copula, and the set of copulas is a [[Semi-group|semi-group]] under the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041080.png" />. The salient fact concerning this operation is the following: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041081.png" /> is a real [[Stochastic process|stochastic process]] with parameter set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041082.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041083.png" /> is the copula of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041085.png" />, then the transition probabilities of the process satisfy the [[Kolmogorov–Chapman equation|Kolmogorov–Chapman equation]] if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041086.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041087.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041088.png" />. 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]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041089.png" /> dimensions. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041090.png" />-copula may be viewed as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041091.png" />-dimensional distribution function whose support is in the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041092.png" />-cube and whose one-dimensional margins are uniform. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041093.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041094.png" />-dimensional distribution function with one-dimensional margins <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041095.png" />, then there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041096.png" />-copula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041097.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041098.png" /></td> </tr></table>
+
$$
 +
H ( x _ {1} \dots x _ {n} ) = C ( F _ {1} ( x _ {1} ) \dots F _ {n} ( x _ {n} ) )
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c11041099.png" />. Moreover, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c110410100.png" />-copula:
+
for all $  ( x _ {1} \dots x _ {n} ) \in \mathbf R  ^ {n} $.  
 +
Moreover, for any $  n $-
 +
copula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c110410101.png" /></td> </tr></table>
+
$$
 +
\max  ( x _ {1} + \dots + x _ {n} - n + 1,0 ) \leq
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c110410102.png" /></td> </tr></table>
+
$$
 +
\leq 
 +
C ( x _ {1} \dots x _ {n} ) \leq  \min  ( x _ {1} \dots x _ {n} ) ;
 +
$$
  
however, while the upper function is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c110410103.png" />-copula for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c110410104.png" />, the lower function is not an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c110410105.png" />-copula for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110410/c110410106.png" />.
+
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.
 
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.

Latest revision as of 17:59, 4 June 2020


A function that links a multi-dimensional 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 [a1] and [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 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) 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) 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 [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 [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 [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 under the operation $ * $. The salient fact concerning this operation is the following: If $ \{ {X _ {t} } : {t \in T } \} $ is a real 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 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) and to a new way of constructing them. It also leads to an interesting area of functional analysis: the study of Markov algebras, [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

[a1] "Advances in probability distributions with given marginals: beyond the copulas" G. Dall'Aglio (ed.) S. Kotz (ed.) G. Salinetti (ed.) , Kluwer Acad. Publ. (1991)
[a2] "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)
[a3] W.F. Darsow, B. Nguyen, E.T. Olsen, "Copulas and Markov processes" Ill. J. Math. , 36 (1992) pp. 600–642
[a4] C. Genest, L.-P. Rivest, "Statistical inference procedures for bivariate Archimedean copulas" J. Amer. Statist. Assoc. , 88 (1993) pp. 1034–1043
How to Cite This Entry:
Copula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Copula&oldid=46484
This article was adapted from an original article by B. SchweizerA. Sklar (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article