multivariate distribution function
A function is said to be a multivariate distribution function if
is non-decreasing in each of its arguments; i.e., for any , the function given by is non-decreasing for any set of such that .
, where is defined as above; i.e., the limit of as is
; i.e. the limit of as each of its arguments approaches infinity, is 1.
Generally, right-continuty of in each of its arguments is added as one of the conditions, but it is not assumed here.
If, in the second condition above, we set for , then is called a (one-dimensional) margin of . Similarly, one defines an -dimensional () margin of by setting of the arguments in to . For each , there are -dimensional margins of . Each -dimensional margin of a multivariate distribution function is itself a multivariate distribution function. A one-dimensional margin is a distribution function.
Multivariate distribution functions are typically found in probability theory, and especially in statistics. An example of a commonly used multivariate distribution function is the multivariate Gaussian distribution function. In , the standard bivariate Gaussian distribution function (with zero mean vector, and the identity matrix as its covariance matrix) is given by
B. Schweizer and A. Sklar have generalized the above definition to include a wider class of functions. The generalization has to do with the weakening of the coordinate-wise non-decreasing condition (first condition above). The attempt here is to study a class of functions that can be used as models for distributions of distances between points in a “probabilistic metric space”.
- 1 B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).
|Title||multivariate distribution function|
|Date of creation||2013-03-22 16:33:50|
|Last modified on||2013-03-22 16:33:50|
|Last modified by||CWoo (3771)|
|Defines||multivariate cumulative distribution function|
|Defines||joint distribution function|