Multivariate Conway-Maxwell-Poisson Distribution: Sarmanov Method and Doubly Intractable Bayesian Inference

ABSTRACT

In this article, a multivariate count distribution with Conway-Maxwell (COM)-Poisson marginals is proposed. To do this, we develop a modification of the Sarmanov method for constructing multivariate distributions. Our multivariate COM-Poisson (MultCOMP) model has desirable features such as (i) it admits a flexible covariance matrix allowing for both negative and positive nondiagonal entries; (ii) it overcomes the limitation of the existing bivariate COM-Poisson distributions in the literature that do not have COM-Poisson marginals; (iii) it allows for the analysis of multivariate counts and is not just limited to bivariate counts. Inferential challenges are presented by the likelihood specification as it depends on a number of intractable normalizing constants involving the model parameters. These obstacles motivate us to propose Bayesian inferential approaches where the resulting doubly intractable posterior is handled with via the noisy exchange algorithm or the Grouped Independence Metropolis–Hastings algorithm. Numerical experiments based on simulations are presented to illustrate the proposed Bayesian approach. We demonstrate the potential of the MultCOMP model through a real data application on the numbers of goals scored by the home and away teams in the English Premier League from 2018 to 2021. Here, our interest is to assess the effect of a lack of crowds during the COVID-19 pandemic on the well-known home team advantage. A MultCOMP model fit shows that there is evidence of a decreased number of goals scored by the home team, not accompanied by a reduced score from the opponent. Hence, our analysis suggests a smaller home team advantage in the absence of crowds, which agrees with the opinion of several football experts. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian inference
• Conway-Maxwell-Poisson distribution
• Exchange algorithm
• Multivariate count data
• Pseudo-marginal Monte Carlo
• Thermodynamic integration

Supplementary Materials

Supplementary materials to Multivariate Conway-Maxwell-Poisson Distribution: Sarmanov Method and Doubly Intractable Bayesian Inference contains additional results to Sections 4 and 5, as well as the additional shunters accident data application.

Data Availability Statement

Datasets and code to fit the proposed model are available as an R package that can be downloaded from https://github.com/luizapiancastelli/mcomp.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

We thank the Editor, Associate Editor, and Referees for their comments and suggestions that led to an improvement of the paper.

Additional information

Funding

L.S.C. Piancastelli and N. Friel wish to acknowledge the financial support of Science Foundation Ireland under grant numbers 18/CRT/6049 and 12/RC/2289 P2. W. Barreto-Souza and H. Ombao would like to acknowledge support by KAUST Research Fund.