is right-continuous at , the lower end point of .
Margin. Note that defined above may or may not exist as each , the upper end point of (). If the limit exists, then we call this limiting function, also written , a (one-dimensional) margin of :
Given an -dimensional rectangle , let’s call each a side of . A vertex of is a point such that each of its coordinates is an end point. Clearly is a convex set and the sides and vertices lie on the boundary of .
-volume. Suppose we have a function , with defined as above. Let be a closed -dimensional rectangle in (), with sides , . The -volume of is the sum
The name is derived from the fact that if , then for each closed rectangle , is the volume of in the traditional sense.
Note, however, depending on the function , may be or even negative. For example, if is a linear function, then the -volume is identically for every closed rectangle , whenever is even. An example where is negative is given by the function , and is the unit square.
-increasing. A function where is an open -dimensional rectange is said to be -increasing if is non-negative evaluated at each closed rectangle .
Any multivariate distribution function is both grounded and -increasing.
A copula, introduced by Sklar, is both a variant and a generalization of a multivariate distribution function.
Formally, a copula is a function from the -dimensional unit cube () to satisfying the following conditions:
every margin of is the identity function.
If we replace the domain by any -dimensional rectangle , then the resulting function is called a subcopula.
For example, the functions , , and defined on the unit cube are all copulas.
(This entry is in the process of being expanded, more to come shortly).
- 1 B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).
|Date of creation||2013-03-22 16:33:43|
|Last modified on||2013-03-22 16:33:43|
|Last modified by||CWoo (3771)|