The Mixture Approach for Simulating Bivariate Distributions With Specified Correlations

Abstract

The Mixture approach is an exact methodology for simulating families of bivariate distributions with specified correlation coefficients, some of which are new. It can accommodate the entire range of correlation coefficients, produces bivariate surfaces that are intuitively appealing, and is often remarkably easy to implement. The approach is introduced in a Bayesian context and demonstrated for the conjugate families of beta and gamma distributions, with special attention given to the bivariate uniform. For these distributions, formulas for correlations have simple closed forms and computations are easy.

KEY WORDS:

• Bayes
• Beta
• Conjugate prior
• Cool
• Exchangeable
• Gamma
• Generating
• Gibbs sampling
• Hierarchical models
• Markov chain Monte Carlo
• Posterior
• Uniform