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Multivariate Distributions with Fixed Marginals and Correlations

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

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 (X1, …, Xn) from a distribution where the marginal of each Xi is specified, as well as the correlation between every pair Xi and Xj. For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between Xi and Xj. Any achievable correlation between Xi and Xj is a convex combination of these bounds. We call the value λ(Xi, Xj) ∈ [0, 1] of this convex combination the convexity parameter of (Xi, Xj) with λ(Xi, Xj) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F1, …, Fn of (X1, …, Xn), we show that λ(Xi, Xj) = λij if and only if there exist symmetric Bernoulli random variables (B1, …, Bn) (that is {0, 1} random variables with mean ½) such that λ(Bi, Bj) = 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
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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