PlanetMath: multivariate distribution function

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multivariate distribution function

(Definition)

A function $F:\mathbb{R}^n\to [0,1]$ is said to be a multivariate distribution function if

is non-decreasing in each of its arguments; i.e., for any $1\le i\le n$ , the function $G_i:\mathbb{R}\to [0,1]$ given by $G_i(x):=F(a_1,\ldots,a_{i-1},x,a_{i+1},\ldots,a_n)$ is non-decreasing for any set of $a_j\in \mathbb{R}$ such that $j\ne i$ .
$G_i(-\infty)=0$ , where is defined as above; i.e., the limit of as $x\to -\infty$ is 0
$F(\infty,\ldots,\infty)=1$ ; 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 $a_j=\infty$ for $j\ne i$ , then is called a (one-dimensional) margin of . Similarly, one defines an -dimensional () margin of by setting of the arguments in to $\infty$ . For each , there are $\binom{n}{m}$ -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 $\mathbb{R}^2$ , the standard bivariate Gaussian distribution function (with zero mean vector, and the identity matrix as its covariance matrix) is given by

$\displaystyle F(x,y)=\frac{1}{2\pi}\int_{-\infty}^x \int_{-\infty}^y \operatorname{exp}\big({-\frac{s^2+t^2}{2}}\big) ds dt$

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”.

Bibliography

1: B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).

"multivariate distribution function" is owned by CWoo.

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See Also: copula

Also defines:

multivariate cumulative distribution function, joint distribution function, margin

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Cross-references: points, distances, class, covariance matrix, identity matrix, mean vector, Gaussian, multivariate Gaussian distribution, statistics, theory, distribution function, infinity, limit, arguments, function
There are 5 references to this entry.

This is version 4 of multivariate distribution function, born on 2007-01-13, modified 2007-01-15.
Object id is 8753, canonical name is MultivariateDistributionFunction.
Accessed 2660 times total.

Classification:

AMS MSC:	60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
	62E10 (Statistics :: Distribution theory :: Characterization and structure theory)

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