copula

copula

Set-up

An

n

-dimensional

rectangle

S

is a subset of

ℝ

n

of the form

I

1

×

⋯

×

I

n

, where each

I

k

is an interval, with

end points

a

k

≤

b

k

∈

ℝ

*

, where

ℝ

*

is the set of extended real numbers (so that

ℝ

itself may be considered as an interval).

Groundedness

. A

function

C

:

S

→

ℝ

is said to be

grounded

if for each

1

≤

k

≤

n

, and each

r

j

∈

I

j

where

j

≠

k

, the function

C

k

:

I

k

→

ℝ

defined by

C

k

⁢

(

x

)

:=

C

⁢

(

r

1

,

…

,

r

j

-

1

,

x

,

r

j

+

1

,

…

,

r

n

)

is right-continuous at

a

k

, the lower end point of

I

k

.

Margin

. Note that

C

k

defined above may or may not exist as each

r

j

→

b

j

, the upper end point of

I

j

(

j

≠

k

). If the limit exists, then we call this limiting function, also written

C

k

, a (one-dimensional)

margin

of

C

:

C

k

⁢

(

x

)

:=

lim

r

j

→

b

j

⁡

C

⁢

(

r

1

,

…

,

r

j

-

1

,

x

,

r

j

+

1

,

…

,

r

n

)

,

where

⁢

j

∈

{

1

,

…

,

n

}

⁢

,

⁢

j

≠

i

.

Given an

n

-dimensional rectangle

S

=

I

1

×

⋯

×

I

n

, let’s call each

I

k

a

side

of

S

. A

vertex

of

S

is a point

v

∈

ℝ

n

such that each of its coordinates is an end point. Clearly

S

is a convex set and the sides and vertices lie on the boundary of

S

.

C

-volume

. Suppose we have a function

C

:

S

→

ℝ

, with

S

defined as above. Let

T

be a closed

n

-dimensional rectangle in

S

(

T

⊆

S

), with sides

J

k

=

[

c

k

,

d

k

]

,

1

≤

k

≤

n

. The

C

-volume

of

T

is the sum

Vol

C

⁡

(

T

)

=

∑

(

-

1

)

n

⁢

(

v

)

⁢

C

⁢

(

v

)

where

v

is a vertex of

T

,

n

⁢

(

v

)

is the number of lower end points that

occur in

the coordinate

representation

of

v

, and the sum is taken over all vertices of

T

.

The name is derived from the fact that if

C

⁢

(

x

1

,

…

,

x

n

)

=

x

1

⁢

⋯

⁢

x

n

, then for each closed rectangle

T

,

Vol

C

⁡

(

T

)

is the volume of

T

in the traditional sense.

Note, however, depending on the function

C

,

Vol

C

⁡

(

T

)

may be

0

or even

negative

. For example, if

C

is a linear function, then the

C

-volume is identically

0

for every closed rectangle

T

, whenever

n

is even. An example where

Vol

C

⁡

(

T

)

is negative is given by the function

C

⁢

(

x

,

y

)

=

-

x

⁢

y

, and

T

is the unit square.

n

-increasing

. A function

C

:

S

→

ℝ

where

S

is an open

n

-dimensional rectange is said to be

n

-increasing

if

Vol

C

is non-negative evaluated at each closed rectangle

T

⊆

S

.

Any

multivariate distribution function

is both grounded and

n

-increasing.

Definition

A copula, introduced by Sklar, is both a variant and a

generalization

of a multivariate distribution function.

Formally, a

copula

is a function

C

from the

n

-dimensional unit cube

I

n

(

I

=

[

0

,

1

]

) to

ℝ

satisfying the following conditions:

1.

C

is

n

-increasing,

2.

C

is grounded,

3.

every margin

C

k

of

C

is the

identity function

.

If we replace the domain by any

n

-dimensional rectangle

S

, then the resulting function is called a

subcopula

.

For example, the functions

C

⁢

(

x

,

y

,

z

)

=

x

⁢

y

⁢

z

,

C

⁢

(

x

,

y

,

z

)

=

min

⁡

(

x

,

y

,

z

)

, and

C

⁢

(

x

,

y

,

z

)

=

max

⁡

(

0

,

(

x

+

y

+

z

-

2

)

)

defined on the unit cube are all copulas.

(This entry is in the process of being expanded, more to come shortly).

References

1

B. Schweizer and A. Sklar,

Probabilistic Metric Spaces

, Dover Publications, (2005).

Title

copula

Canonical

name

Copula

Date of creation

2013-03-22 16:33:43

Last modified on

2013-03-22 16:33:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62A01

Classification

msc 54E70

Related topic

MultivariateDistributionFunction

Related topic

ThinSquare

Defines

subcopula

Defines

n

-increasing

Defines

grounded

Defines

margin

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