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'multivariate distribution function'
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| Title of object: |
multivariate distribution function |
| Canonical Name: |
MultivariateDistributionFunction |
| Type: |
Definition |
| Created on: |
2007-01-13 17:59:26 |
| Modified on: |
2007-01-14 23:56:49 |
| Classification: |
msc:60E05, msc:62E10 |
| Defines: |
multivariate cumulative distribution function, joint distribution function, margin |
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Content:
A function $F:\mathbb{R}^n\to [0,1]$ is said to be a \emph{multivariate distribution function} if
\begin{enumerate}
\item $F$ 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 $j\ne i$.
\item $G_i(-\infty)=0$, where $G_i$ is defined as above; i.e., the limit of $G_i$ as $x\to -\infty$ is $0$
\item $F(\infty,\ldots,\infty)=1$; i.e. the limit of $F$ as each of its arguments approaches infinity, is 1.
\end{enumerate}
Generally, right-continuty of $F$ 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 $a_1=\cdots=a_{i-1}=a_{i+1}=\cdots= a_n$, then $G_i(x)$ is called a one-dimensional \emph{margin} of $F$. Similarly, one defines an $m$-dimensional ($m<n$) \emph{margin} of $F$ by setting $n-m$ of the arguments in $F$ to $\infty$. For each $m<n$, there are $\binom{n}{m}$ $m$-dimensional margins of $F$. Each $m$-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
$$F(x,y)=\frac{1}{2\pi}\int_{-\infty}^x \int_{-\infty}^y \operatorname{exp}\big({-\frac{s^2+t^2}{2}}\big) ds dt$$
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''.
\begin{thebibliography}{8}
\bibitem{bs as} B. Schweizer and A. Sklar, {\em Probabilistic Metric Spaces}, Dover Publications, (2005).
\end{thebibliography} |
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