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CPS1823 Ishapathik D. et al.
since incorporating dependence for discrete data demands more
sophisticated techniques.
When we have a data with small counts or several zero counts, the normal
approximation is not adequate and multivariate Poisson models would be
more appropriate. Karlis and Meligkotsidou (2005) gave a method to construct
a multivariate Poisson random vector =(1,2,…,) with equi-correlation
structure, and this can be described by the stochastic representation
1 =1 +0,2 =2 +0,…, = +0

where 1,2,…, and 0 are +1 independent Poisson random variables with
means 1,…, and 0. The counts 1,2,…, are dependent and the covariance
matrix of is given by
… 0
0
1
…
( 0 2 0 )
…
…
0
0
which is completely specified by the parameters 0,2,3,…, of the
distribution. This simplistic model has several limitations even in the bivariate
case because of the structure. Several authors have proposed variations of the
bivariate Poisson to model correlated bivariate count data with inflated zero
observations. For situations where counts of (0,0) are inflated beyond
expectation, the proposed zero-inflated bivariate Poisson distribution by
Maher (1982) is appropriate. Sen et al. (2018) introduced two bivariate Poisson
distributions that would account for inflated counts in both the (0,0) and (,)
cells for some >0. The two bivariate doubly inflated Poisson distributions
(BDIP) are parametric models determined either by four (,1,2,3) or five
(1,2,1,2,3) parameters. The authors have discussed distributional properties
of the BDIP model such as identifiability, moments, conditional distributions,
and stochastic representations, and provide two methods for parameter
estimation. Wang (2017) built an alternative tractable statistical model. This
model which uses the Gaussian copula extends easily to the multivariate
doubly inflated Poisson distribution. Maximum likelihood estimation of the
parameters for this copula based model was also studied in Wang (2017). In
this paper we further extend the Gaussian copula model of Wang (2017) to the
regression set up in the presence of covariates.

2. Doubly-inflated Multivariate Poisson Distribution
In this section we will discuss an extension of the multivariate Poisson
distribution that accounts for the inflated frequencies at zero and some other
value =[1,2,…,]′. This extension known as the doubly-inflated multivariate
Poisson distribution is constructed using a latent variable having the PMF

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