Distribution and Simulation of Random Correlation Matrices: Hyperspherical Parameterization of the Cholesky Factor

• Activity Number: 248
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Computing
• Author(s): Xiao Wang*+ and Mohsen Pourahmadi
• Companies: Texas A&M and Texas A&M
• Keywords: High-dimensional correlation matrix ; Hyperspherical Parameterization ; Cholesky Factors

Abstract

The distribution of a random correlation matrix is useful in finding the distributions of its entries, eigenvalues, condition numbers and norms among others. However, finding distribution of random correlation matrices constructed from existing methods such as random eigenvalues and random Gram matrices has proved elusive. We study distribution of random correlation matrices using the hyperspherical parameterization of their Cholesky factors and the distributions of the related angles. The latter is chosen in such a way that the distribution of the correlation matrix is proportional to its determinant and hence independent of the labeling or order of the variables. We highlight the roles of this procedure in generating high-dimensional correlation matrices with specific distributions and guaranteed positive-definiteness, and modeling correlation matrices using covariates.