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CPS1823 Ishapathik D. et al.

distributed as DIMP(1,2,1,2,…,,), where 1 ≡1() , 2 ≡2() and
≡() for =1,2,… ; =1,2… . The mean vector of this distribution is
()=[1(),2(),…,()]′
where
() = 2() +[1−1()−2()](), (3.1)
as given in equation (2.4). For relating the mean vector (), we use the log
link function given by
log[()] = (), (3.2)

or equivalently

()= exp [ ( )]− 2 ( ) , (3.3)
[1− 1 ( )− 2 ( )]
where 1()+2()<1 for =1,2,…,. Further we assume ()’s are linear
functions of the covariates,
1()=′1()1
2()=′2()2
…
()=′(), (3.4)
where 1,2,…, are vector functions of , and 1,2,…, are unknown
regression parameter vectors of order 1 ×1, 2 ×1, …, ×1 respectively. Let us
denote the combined regression parameter vector as =[1′,2′,…,′]′ of order
×1 with =1 +2 +⋯+. Our regression problem involves estimating the
regression parameter from the observed data. For estimating 1() and 2()
from the data, we define a latent variable () given by
2 ify = 0,
i
( ) ={2 ify = k,

i
0 otherwise.
Now, we fit the multivariate logit model given by

()=[()=2] = exp [ 1 ( )] ,
1
1+exp[ 1 ( )]+exp [ 2 ( )]
()=[()=1] = exp [ 2 ( )] , (3.5)
2
1+exp[ 1 ( )]+exp [ 2 ( )]

so that 1()+2()<1, where 1()=1′()1 and 2()=2′()2 are the
corresponding linear predictors of the model with unknown parameter vectors
1, and 2 of order 1 and 2 respectively and 1(⋅) and 2(⋅) are the
corresponding vector functions of . We denote the combine vector as
=[1′,2′]′ of order =1 +2. In the next section, we discuss maximum likelihood

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