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Multivariate Distributions with Fixed Marginals and Correlations

Part of:Algorithms - Computer ScienceProbabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 January 2018

Mark Huber and
Nevena Marić
Mark Huber*
Affiliation:
Claremont McKenna College
Nevena Marić*
Affiliation:
University of Missouri - St. Louis
*
∗ Postal address: Claremont McKenna College, 850 Columbia Avenue, Claremont, CA 91711, USA. Email address: autotomic@gmail.com
∗∗ Postal address: University of Missouri - St. Louis, 1 University Boulevard, St. Louis, MO 63121, USA.
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Abstract

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Consider the problem of drawing random variates (X 1, …, X n ) from a distribution where the marginal of each X i is specified, as well as the correlation between every pair X i and X j . For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between X i and X j . Any achievable correlation between X i and X j is a convex combination of these bounds. We call the value λ(X i , X j ) ∈ [0, 1] of this convex combination the convexity parameter of (X i , X j ) with λ(X i , X j ) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F 1, …, F n of (X 1, …, X n ), we show that λ(X i , X j ) = λij if and only if there exist symmetric Bernoulli random variables (B 1, …, B n ) (that is {0, 1} random variables with mean ½) such that λ(B i , B j ) = λij . In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.

Keywords

MultivariatecopulacorrelationMonte Carlo algorithm

MSC classification

Primary:65C05: Monte Carlo methods
Secondary:68W20: Randomized algorithms

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015, pp. 602 - 608
Copyright
© Applied Probability Trust 

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