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CPS1823 Ishapathik D. et al.
ifz = 2,
1
ifz = 2,
() = { 2 (2.1)

1 − − ifz = 2,
2
1
0 elsewhere,
where , ∈ (0,1) with 1 +2 <1. Using the properties of joint and
1
2
conditional probability we arrive at the final form of the marginal PMF of Y.
The marginal PMF of is then given by;
() = ∑ (, )
,

+ (1 − − ) () ify = [0,0, … ,0]′

1
1
2
={ + (1 − − ) () ify = [ , , … , ]′ (2.2)

2
1
2

2
1
(1 − − ) () elsewhere.
1
2

The distribution () is known as a doubly inflated multivariate Poisson
distribution (DIMP) with parameters (1,2,1,2,…,,) , where 0<1 <1 +2 <1
and >0 for = 1,2,…,. If the correlation matrix is a structured matrix (),
we will write as the parameter instead of . The marginal distribution of the
th component of can be obtained as
( ) = ∑ … ∑ ∑ … ∑ ()

1 −1 +1
+ (1 − − ) ( ) ifyj = 0,

2
1
1

= { + (1 − − ) ( ) ifyj = (2.3)
2

1

2
(1 − − ) ( ) elsewhere.
1

2
where () is the PMF of Poisson with mean . Using (2.3), we can see that
=()
=2 +(1−1 −2), (2.4)
and this reduces to when 1 and 2 converge to zero.

3. Regression with DIMP Distribution
Here we discuss performing regression for multivariate count data that
has inflated frequencies at a positive count besides zero. To set up the
regression, let us assume that associated with the th multivariate count
response =[1,2,…,]′ there is a vector of covariates = [1,2,…,]′
observed. The goal of the regression is to study the relation between and
. Assuming that the sample data consists of inflated frequencies for count
zero and another count . In this case it is reasonable to assume that is
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