On the three-dimensional negative binomial distribution

ABSTRACT

We consider a particular generalization of the negative binomial distribution to the multivariate case obtained through a specification of the probability generating function as the negative power of a certain polynomial. The probability function itself has previously been derived for the two-dimensional case only, and inference in the multivariate negative binomial distribution has been restricted to the use of composite likelihood based on one- or two-dimensional marginals. In this article, we derive the three-dimensional probability function as a sum with all terms positive and study the range of possible parameter values. We illustrate the use of the three-dimensional distribution for modeling three correlated SAR images.

KEYWORDS:

• Correlated count data
• Inference
• Mixture model
• Multivariate analysis
• Image analysis

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 62H05
• Secondary: 62H12

Acknowledgments

We would like to thank Ege Rubak for fruitful discussions and CNES Toulouse for letting us use the volcano data in Sec. 3.