Easily simulated multivariate binary distributions with given positive and negative correlations

We consider the problem of defining a multivariate distribution of binary variables, with given first two moments, from which values can be easily simulated. Oman and Zucker [Oman, S.D., Zucker, D.M., 2001. Modelling and generating correlated binary variables. Biometrika 88, 287–290] have done this when the correlation matrix of the binary variables is the Schur product of a parametric correlation matrix C appropriate for normal variables (intraclass, moving average or autoregressive), having non-negative entries, with a matrix whose entries comprise the Fréchet upper bounds on the pairwise correlations of the binary variables. We extend their method to include negative correlations; moreover, we extend the range of positive correlations allowed in the moving-average case. We present algorithms for simulation of data from these distributions, and examine the ranges of correlations obtained.