  In this paper, 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
non-diagonal 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 a Bayesian inferential approach where the
resulting doubly-intractable posterior is dealt with via the exchange algorithm
and the Grouped Independence Metropolis-Hastings algorithm. Numerical
experiments based on simulations are presented to illustrate the proposed
Bayesian approach. We analyze 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 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.
