Multivariate distributions with fixed marginals and correlations

Huber, M. E.; Maric, Nevena

doi:10.48550/arxiv.1311.2002

Cited by 3 publications

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“…Introduction of the concurrence vector has its motivation from the context of symmetric Bernoulli random variables [2]. Let B 1 , .…”

Section: Cut Polytopesmentioning

confidence: 99%

“…It s well known that every correlation matrix belongs to E n , the set of symmetric positive semi-definite matrices with all diagonal elements equal to 1. For Gaussian marginals, the entirety of E n can be realized, but surprisingly enough, for other common distributions very little is known [2]. For multivariate symmetric Bernoulli the problem was recently solved in [3] where the polytope R(B n ) was characterized by identifying its vertices.…”

Section: Introductionmentioning

confidence: 99%

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Cut polytope has vertices on a line

Maric

2018

Electronic Notes in Discrete Mathematics

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The cut polytope CUT(n) is the convex hull of the cut vectors in a complete graph with vertex set {1, . . . , n}. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation with admissible correlations of symmetric Bernoulli random variables. That probabilistic interpretation is a starting point of this work in conjunction with a natural binary encoding of the CUT(n). We show that for any n, with appropriate scaling, all encoded vertices of the polytope 1-CUT(n) are approximately on the line y = x − 1/2.

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“…Introduction of the concurrence vector has its motivation from the context of symmetric Bernoulli random variables [2]. Let B 1 , .…”

Section: Cut Polytopesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cut polytope has vertices on a line

Maric

2018

Electronic Notes in Discrete Mathematics

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“…The second reason comes from Huber and Marić [15] where this distribution was shown to be in a certain sense the most difficult problem: for general marginals and some correlations it is possible to transform the problem into symmetric Bernoulli marginals.…”

Section: Introductionmentioning

confidence: 99%

“…Consider a matrix Σ that is potentially a correlation matrix for a particular choice of marginal distributions. Then what the method of [15] does is build a second correlation matrix Σ B such that if is possible to have a multivariate distribution with symmetric Bernoulli marginals and correlation Σ B , then it is possible to build a multivariate distribution with the original marginals and correlation matrix Σ.…”

Section: Introductionmentioning

confidence: 99%

Bernoulli Correlations and Cut Polytopes

Huber¹,

Maric²

2017

Preprint

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Given n symmetric Bernoulli variables, what can be said about their correlation matrix viewed as a vector? We show that the set of those vectors R(Bn) is a polytope and identify its vertices. Those extreme points correspond to correlation vectors associated to the discrete uniform distributions on diagonals of the cube [0, 1] n . We also show that the polytope is affinely isomorphic to a well-known cut polytope CUT(n) which is defined as a convex hull of the cut vectors in a complete graph with vertex set {1, . . . , n}. The isomorphism is obtained explicitly as R(Bn) = 1 − 2 CUT(n). As a corollary of this work, it is straightforward using linear programming to determine if a particular correlation matrix is realizable or not. Furthermore, a sampling method for multivariate symmetric Bernoullis with given correlation is obtained. In some cases the method can also be used for general, not exclusively Bernoulli, marginals.

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“…In Dukic and Marić (2013) (and see also Huber and Marić, 2015), it is shown how to simulate a bivariate Geometric distribution that attains any value between the maximum and minimum correlation, although these methods require knowledge of the maximum and minimum correlation.…”

Section: Introductionmentioning

confidence: 99%

Minimum correlation for any bivariate Geometric distribution

Huber,

Maric

2014

Preprint

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Consider a bivariate Geometric random variable where the first component has parameter p 1 and the second parameter p 2 . It is not possible to make the correlation between the marginals equal to -1. Here the properties of this minimum correlation are studied both numerically and analytically. It is shown that the minimum correlation can be computed exactly in timeOne method for generating a bivariate geometric with target correlation requires computing this minimum correlation. The minimum correlation is shown to be nonmonotonic in p 1 and p 2 , moreover, the partial derivatives are not continuous. For p 1 = p 2 , these discontinuities are characterized completely and shown to lie near (1 -roots of 1/2). In addition, we construct analytical bounds on the minimum correlation.

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Multivariate distributions with fixed marginals and correlations

Cited by 3 publications

References 0 publications

Cut polytope has vertices on a line

Cut polytope has vertices on a line

Bernoulli Correlations and Cut Polytopes

Minimum correlation for any bivariate Geometric distribution

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