Maximum likelihood estimation of Gaussian copula models for geostatistical count data

Abstract

This work investigates the computation of maximum likelihood estimators in Gaussian copula models for geostatistical count data. This is a computationally challenging task because the likelihood function is only expressible as a high dimensional multivariate normal integral. Two previously proposed Monte Carlo methods are reviewed, the Genz–Bretz and Geweke–Hajivassiliou–Keane simulators, and a new method is investigated. The new method is based on the so–called data cloning algorithm, which uses Markov chain Monte Carlo algorithms to approximate maximum likelihood estimators and their (asymptotic) variances in models with computationally challenging likelihoods. A simulation study is carried out to compare the statistical and computational efficiencies of the three methods. It is found that the three methods have similar statistical properties, but the Geweke–Hajivassiliou–Keane simulator requires the least computational effort. Hence, this is the method we recommend. A data analysis of Lansing Woods tree counts is used to illustrate the methods.

Keywords:

• Data cloning
• Gaussian random field
• Markov chain Monte Carlo
• Multivariate normal integral
• Simulated likelihood

AMS SUBJECT CLASSIFICATIONS:

• 60G10
• 60G60
• 62M30

Acknowledgements

We warmly thank two anonymous referees for helpful comments and suggestions that lead to an improved article.

Notes

1 With the convention that ${\sum }_{j=1}^0{v}_i=0$.

Additional information

Funding

This work was partially supported by the U.S. National Science Foundation Grant DMS–1208896, and received computational support from Computational System Biology Core, funded by the National Institute on Minority Health and Health Disparities (G12MD007591) from the National Institutes of Health.