Multivariate distributions with fixed marginals and correlations

Abstract

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 ) = λ _ij 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 ) = λ _ij . In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.