Applied Mathematics, 2011, 2, 824-829
doi:10.4236/am.2011.27110 Published Online July 2011 (http://www.SciRP.org/journal/am)
Bivariate Zero-Inflated Power Series Distribution
Patil Maruti Krishna1, Shirke Digambar Tukaram2
1Department of Stat i s t i c s, P. V. P. Mahavidyalaya, Kavathe Mahankal, Sangli, India
2Department of Stat i s t i c s, Shivaji University, Kol h apur, India
Received December 19, 2010; revised May 12, 2011; accepted M ay 15, 2011
Abstract
Many researchers have discussed zero-inflated univariate distributions. These univariate models are not
suitable, for modeling events that involve different types of counts or defects. To model several types of de-
fects, multivariate Poisson model is one of the appropriate models. This can further be modified to incorpo-
rate inflation at zero and we can have multivariate zero-inflated Poisson distribution. In the present article,
we introduce a new Bivariate Zero Inflated Power Series Distribution and discuss inference related to the
parameters involved in the model. We also discuss the inference related to Bivariate Zero Inflated Poisson
Distribution. The model has been applied to a real life data. Extension to k-variate zero inflated power series
distribution is also discussed.
Keywords: Bivariate Zero-Inflated Power Series Distribution, Bivariate Zero-Inflated Poisson Distribution,
K-Variate Zero-Inflated Power Series Distribution
1. Introduction
In a manufacturing process there may exist several types
of (say m) defects—for example, solder short circuits,
solder voids, absence of solder etc. on one printed circuit
board. These defects cause different types of product
failure and generate different types of equipment prob-
lems. In the above example there can be only one type of
defect which occurs more frequently and the other de-
fects occurs very rarely. Another situation could be both
types of defects occur rarely and so on. To model several
types of defects, multivariate Poisson model is one of the
appropriate models to use. This can further be modified
to incorporate inflation at zero and we can have multi-
variate zero-inflated Poisson (MZIP) distribution. There
are several ways to construct MZIP distributions. In the
literature, Chin-Shang et al. [1] have discussed various
types of MZIP models and investigated their distribu-
tional properties. Deshmukh and Kasture [2] have stud-
ied bivariate distribution with truncated Poisson marginal
distributions. Gupta et al. [3] have considered inflated
distributions at the point zero and studied the structural
properties of the inflated distribution. Gupta et al. [4]
have discussed score test for zero-inflated generalized
Poisson regression model. Holgate [5] described the es-
timation of covariance parameter of bivariate Poisson
distribution by iterative method. Lambert [6] considered
zero-inflated Poisson regression model. Laxminarayana
et al. [7] have studied bivariate Poisson distribution and
the distributional properties of the model. Patil and
Shirke [8] studied testing parameter of the power series
distribution of a zero-inflated power series model. Patil
and Shirke [9] also studied equality of inflation parame-
ters of two zero-inflated power series distributions. It
appears that majority of the study in the literature is re-
stricted to Poisson distribution and its extension to mul-
tivariate set up. Relatively less has been reported for the
family of distributions containing other distributions.
In this article, we introduce a new Bivariate Zero-In-
flated Power Series Distribution (BZIPSD) and discuss
inference related to the parameters involved in the same.
The rest of the paper is organized as follows. Section 2,
introduces the BZIPSD along with moments of the same.
Section 3, deals with inference related to the parameters
involved in the BZIPSD. In Section 4, we discuss infer-
ence related to Bivariate Zero-Inflated Poisson Distribu-
tion (BZIPD). The data set reported by Arbous and Ker-
rich [10] is modeled by Bivariate Zero Inflated Poisson
Distribution. The paper concludes with generalization to
multivariate setup.

P. M. KRISHNA ET AL.825
2. Bivariate Zero-Inflated Power Series
Distribution
Let
X
and Y be two random variables with probabil-
ity mass functions


1
11
11
,
x
ax
Px f



 and



2
22
22
,
y
by
Py f




,.
x
yT
where is the common support of T
X
and Y, 10,


20

, , (.)a(.) 0b
 
11 1
x
fax

 ,

22

2
.
y
fby

 
Define,



 
  

  

,121122
112 2
,, ,,,,
1
XY
Pxy PxPy
gx EgXgy EgY
 




 


, (2.1)
where,


1
g
x and 2()
g
y are bounded function on 2

.
note that We


12
, ,
,
,
XY
Pxy,


is a proper bivariate
distribution for a suitable choice of

. Based on the
distribution (2.1), in the followingntroduce three
types of BZIPSD.
Type-I BZIPSD
we i
: When there is an inflation only at

x

,0,0y


, we define the BZIPSD as

 

,12
12
,12
1ππ 0, 0,,,,,0, 0, 0π1
,,π,, ,
π,,π,, ,,,0,0
XY
XY
Pxy
Pxy Pxy xy




 




(2.2)
Type-II BZIPSD: When there is inflation at
X
component only, we define the BZIPSD as
(2.3)
Type-III BZIPSD: When there is inflation at component only, we define the BZIPSD as
(2.4)
In the present discussion we focus only on Type-IBZIPSD,
re
Moment Generating Function
of (X,Y) is

 

,12
12
,12
1ππ 0,,,,,,0,0,0π1
,,π,, ,
π,,π,,, ,,0,0
XY
XY
Py xy
Pxy Pxyxy




 




Y-

 

,12
12
,12
1ππ ,0,,,,,0,0, 0,1,2,
,,π,,,
π,,π,,,,0,1, 2,,1, 2, 3,
0π1
XY
XY
Pxxy xx
Pxy Pxyx y

 

 







sults on the remaining two can be obtained analogously. The moment generating function



 


 
 
12
12
,12
,12
1122111122 22
,
,
1ππ
tX tY
XY
XY
tX tY
MttEe
Mtt
MtMtEegX MtEgXEegY MtEgY



 
 




 
(2.5)
Therefore, we have

Mt M
 
 
 
 
1,1
11
2,2
22
,0
1ππ
0,
1ππ
XXY
YXY
t
M
t
Mt Mt
M
t
 

 

(2.6)
where

1X
M
t


j
f


and

j
f





j
f




denote
and

2Y
M
t
m vari
are the moment generat-
ing funf randoables ctions o
X
and Y of zero-
inflated power series distributionnd a


11
M
t and

22
M
t are the moment generating functions dom
s having power series distribution with parame-
ters 1
of ran
variable

and 2

respectively.
Suose pp and


2
2
j
f



ives us
respectively for This g
 
1, 2j.

11 1
1
f
EX Mf1
π
0
X






 

22 2
22
π
0
Y
f
EY Mf






   

2
111
1
11 111
11 11
π
() f Var Xff
ff

  



  




P. M. KRISHNA ET AL. 826
   

2
22 2
2
22 222
22 22
π
() f Var Xff
ff

  



 




,
and the correlation coefficient is
 

  
 
   
1111 11222222
112 2
12
22
11 2 21112 22
111112 2222
112 2
ππ
fefeff efef
efe f
ff ff
ffff
ff



   



 


 
  
 


 

 

   
 

 

(2.7)
3. Estimation of the Parameters of BZIPSD
Let ,



,1,2,3,
ii
X
Yi n be a random sample ob-
served from BZIPSD

12
π,,, .

 The likelihood
functe is given by. ion for the observed random sampl

  
 
12
1
π,,,;,,
i
a
n
Lxy 

 (3
12
1
12
1ππ 0, 0, ,,
π,,,, i
XY
i
a
XY i i
P
Px y





.1)
where if 0
and otherwise.
The codinunctven by,
1
i
a
rrespon
 
,0,
ii
xy 
g log likelihood f
0
i
a
ion is gi

  
12
012
lo ,
log 1 ππ 0,0,,,
XY
nP



,1
2
11
gπ, ,;,,
log πlog, , ,,
nn
ii XYii
ii
Lxy
aaPxy






)  (3.2
log 0
π
L
,
1
log 0 L


,
2
log 0 L


 and log 0
L




give the following equations.



0, 12
0, 0,,,
(1 π)π0, 0,,
XY a
nP
P



1
,1 2
10
,π
n
i
i
XY




(3.3)


 



1
1
0, 12
,12
,12
1,12
π0, 0,,,
1ππ 0, 0,,,
,,,, 0
,,,,
XY
XY
nXYi i
i
iXYi i
nP
P
Pxy
aPxy









 





(3.4)






2
2
0, 12
,12
,12
1,12
π0, 0,,,
1ππ (0,0,,, )
,,,, 0
,,,,
XY
XY
nXYi i
i
iXYi i
nP
P
Pxy
aPxy









 









0, 12
,12
,12
1,12
π0, 0,,,
1ππ (0,0,,, )
,,,, 0
,,,,
XY
XY
nXYi i
i
iXYi i
nP
P
Pxy
aPxy















(3.6)
where denote

,(.) XY
P

,(.)
XY
P



Solving Equations (3.3) to (3.6) simultaneously we get
maximum likelihood estimators of the desired four pa-
rameters. We note that all the four likelihood equations
are non-linear in nature and do not have clo
lution. Now, we discuss a particular case
namely BZIPD.
sed form so-
of BZIPSD
4. Bivariate Zero-Inflated Poisson
Distribution
Let us set




1
!ax x

,
 
1
!by y ,


1
x
g
xe

,


2
y
g
ye

,


1
11
fe



,

2
22
fe



 in the model
(2.1). Then we get BZIPD with probability mass func-
tion.




(3.5)








12
1
12
,
,
,0,0
π1,
,0,0
where 11
xy c
xy
P
12 12
1ππ 111
cc
2
()
c
!!
XxYy
xy
eee e
xy
ce

 


ee
e
 

 
e
   

xy


 


(4. 1
The moment generating function of (X,Y) is
(4.2)
  





)




 











12
1122
00
112 2
1ππ ,,
1.
txt y
xy
ePxePy
gxEgXgyEgY





 


P. M. KRISHNA ET AL.827
It is clear from the expressions of moment generating
functions of
X
and that the marginal distributions
of X and Y are univariate zero-inflated power series dis-
tributions with paramters
Y
e

1
π,

and

2
π,

respec-
tively. Further we have
 
12
π and πEX EY

 
 

  
11
22
π11π
and π11π
Var X
Var Y




The correlation coefficient is turns out to be




12
2c


12
2
12 12
11π1π
ce




 
(4.3)
Remark 1: When there is no inflation , the cor-


π1
relation coefficient

is given by

12
2
12
c
ce

 

,
which coincides with the correlation coefficient given by
Laxminarayan et al. [8]
Remark 2: If we choose (.)
g
to be anher suit-
able bounded function, we will have different form of
BZIPD. Some other possible functions can be


,
z
g
za 01a;

 ()
z
g
ze


0
etc.
Remark 3: If


, we get Bivariate Zero-Inflated
Poisson distribution based on two independent random
variables.
Estimation of the Parameters of BZIPD
Suppose


,;1,2,,
ii
x
yi n is a random sample ob-
served from BZIPD


,;1,2, ,i
12
π,, n

 . Then the
likelihood function is given by






 
121 2
12
12
12
1
()
1
12
π,,,;,
(1) π111
π1
!!
i
i
ii ii
na
cc
i
a
xy xy
cc
ii
Lxy
eee
eee ee
xy
 




 


 


 

 

 




(4.4)
where 1
i
a

if




,0,
ii
xy  y ot 0 and otherwise.
The corresponding log likelihood is given by,
0
i
a












  
12
11 1
loglog log
ii ii i
ii i
ax ay a 
 
 




12 12
12
12 012
11
11
log π,, ,;,log1ππ 11 1log(π)
!log!log 1ii
nn
cc ii
ii
nnnnn xy
cc
ii ii
ii
Lxyn eeaa
xayae eee
 

 

 








(4.5)
The mles of the parameters can be obtained
equations
e
by solving
and
log 0 L




in the followi
simultaneously. These equations are given
ng:
log 0
π
L
,
12
log 0 L




log 0 L


,

 







12 12
12
1ππ 11e


 12
01
11 1
0
π
1
n
cc i
i
cc
a
e
ee
 


 


 1ne e


 

(4.6)
 
 




 



122 112
12 12
12
12
()
0
1
0
1
1i
π11 11
1ππ 11 1
0
1
i
ii
c c
cc
n
y cc
ii n
iixy
cc
nec eeee
eee
ax eee
nnc aeeee
 
 






 
 
 

 

 



 

 


cc    
(4.7)
 

 






121 21
11
c
e e
  

2
12 12
21
12
()
0
1
0
1
2
π11
1ππ 11 1
0
1
i
ii
cc c
cc
n
x cc
ii n
iixy
cc
i
nec ee
eee
ay eee
nnc aee ee
 
 






 
 
 

 


 



 

 


 
(4.8)

P. M. KRISHNA ET AL. 828







1212
12 12
12
12
()
0
1
π11
1ππ 111
0
1
ii
ii
cc
cc
xy cc
n
ixy
cc
i
nee e
eee
ee ee
aee ee
 
 





 
 




 





 


of the marginal distribution of Y to ZIPD we get Chi
square statistic = 0.6065 and P value = 0.4360. The table
value of = 3.841. Therefore, ZIPD fits well for
X and Y dat
Thus now we can test whet
BZIPD
(4.9)
From the above equations, it is clear that Equations
(4.6) to (4.9) are non-linear in nature. Solving these
equations is computationally cumbersome. Laxminara-
yan et al. [7], adopt method of moments for the model
without inflation parameter (i.e. ). In their model
they have used estimates based on Method of Moment
Estimators (MME), which coincide with Maximum
Likelihood Estimators (MLE) of the marginal distribu-
tions. This is not the case for the joint distribution. We
have to solve four equations simultaneously in order to
et the MLEs. In the following we obtain maximum like-
lihood estimators for the following example and test for
goodness of fit.
5. An Application
The data set in Table 1 reported by Arbous and Kerrich
[10],
π1
g
represents accidents sustained by 122 railway men
in consecutive periods of 6 and 5 years.
X is the accident distribution of 122 railway men dur-
ing 1937-1942 and Y is the accident distribution of 122
railway men during 1943-1947.
By assuming marginal distributions of X is ZIPD

1
π,.

The MLEs of X data are ˆ π0.8938 and
1
ˆ1.2564
  Similarly assuming marginal distribution of
Y is ZIPD

2
π,.

The MLEs of Y data are ˆ π0.8494

,
2
ˆ1.3221
 . Using these mles we fit the data of the mar-
ginal distribution of X to ZIPD, we get Chi square statis-
tic = 0.74843 and P value = 0.3869. If we fit the data
Table 1. Bivariate accident distribution of 122 railway men
0 1 2 3 4 5 6 7 Total
during two periods.
Y
X
0 21 14 8 1 44
1 17 12 8 3 1 1 42
9
5
2 6 9 2 2 2 21
3 1 1 3 3 1
4 1 3 4
2 2
6
7
Total 46 39 21 11 4 1 122
2
(1,0.05)

a.
her data is coming from


12
π,, .


n (4.5) using
m likeliho
0
Maximizing the log likelihood in the
Equatio MATLAB R12 software we get
maximuod estimators of the parameters as
π0.94

, 11.210


, 21.20

, 1.22 0


-Inflated
. With
ZeroPoisson
Distribution to the above data. The expected frequencies
are as shown in the Table 2.
From the chi-square good
calculated , is less than the table value of
alue is 0.392369. Hence we
o-Inflated Poisson Distribu-
tion fits well for the data.
Remark 4: There can be mny ways to define k-vari-
A k-variate Zero-Inflated Power Series Distribution
cabeed
these parameters we fit Bivariate
ness of fit, we observed that
24.102062
 
9.488 . The P v
hat Bivariate Zer
2
(4,0.05)
 
conclude t
a
ate ZIPSD by extending the above defined BZIPSD. One
of the ways is given below.
n e dfinas




1ππ ,,f 0
,,f
X
Px
Px

i
,π
πP,x i x0
Xx


 




where


12
,,,
k



,

12
,, k
X
XX X 



  

,,
,,
i
X
Xii
Px
Pxg EgX








i
11i
kk
ii
x


,
i
Xii
Px

ries Distriuti
is probability ma fucti ofowr Se-
o
Inted to parameters involved in
man apsiarly.
modat
ons. Further work under consideration is
te
model fo
≥2 Total
ssnon Pe
b n.
ference
odel c
rela
be
the
ted
this
ttem mil
In the present work we introduced a new bivariate
zero-inflated power series distribution. This distribution
can accome number of zero-inflated bivariate dis-
crete distributi
sting of independence for BZIPSD. Application of the
proposedr some other distributions like Bivari-
ate Zero-Inflated Negative Binomial Distribution or
k-variate zero inflated Poisson distribution can also be
Table 2. Expected frequencie s using BZIPD.
Y
X0 1
0 5500 8.7747 41.5161 21.1914 11.
1 11.6746 15.1344 14.5680 41.3770
≥2 8.9933 14.7627 15.3421 39.0981
Total 41.8593 41.4471 38.6848 121.9912

P. M. KRISHNA ET AL.829
[1] L. Chin-Shang, K. Kyungmoo, J. P. Peterson and P. A.
Brinkley, “Multivariate Zero-Inflated Poisson
and Their Applications,” Technometrics, Vol. 41, No. 1,
99, pp. 29-38. doi:10.2307/1270992
doi: 10.1081/STA-120026576
considered. These models are useful to model zero-in-
flated bivariate data.
[5] P. Holgate, “Estimation for the Bivariate Poisson Distri-
307/1269547
6. References
Models
19
[2] S. R. Deshmukh and M. S. Kasture, “Bivariat
tion with Truncated Poisson Marginal Distributions,”
Communication in Statistics: Theory and Metord
31, No. 4, 2002, pp. 527-534.
doi:10.1081/STA-120003132
e Distribu-
s, Vol.
[3] P. L. Gupta and R. C. Tripathi, “Inflated Modified Power
Series Distributions with Applications,” Communication
in Statistics: Theory and Metords, Vol. 24, No. 9, 1995,
pp. 2355-2374. doi:10.1080/03610929508831621
[4] R. L. Gupta and R. C. Tripathi “Score Test for Zero-
Inflated Generalized Poisson Regression Model,” Com-
munication in Statistics: Theory and Metords, Vol. 33,
No. 1, 2004, pp. 47-64.
bution,” Biometrika, Vol. 51, No. 1-2, 1964, pp. 241-245.
[6] D. Lambert, “Zero-Inflated Poisson Regression, with an
Application to Defects in Manufacturing,” Technometrics,
Vol. 34, No. 1, 1992, pp. 1-14. doi:10.2
[7] J. Lakshiminarayana, S. N. N. Pandit and K. Srinivasa
Rao, “On a Bivariate Poisson distribution,” Communica-
tion in Statistics: Theory and Metords, Vol. 28, No. 2,
1999, pp. 267-276.
[8] M. K. Patil and D. T. Shirke, “Testing Parameter of the
Power Series Distribution of a Zero-Inflated Power Series
Model,” Statistical Methodology, Vol. 4, No. 4, 2007, pp.
393-406. doi:10.1016/j.stamet.2006.12.001
[9] M. K. Patil and D. T. Shirke, “Tests for Equality of Infla-
tion Parameters of Two Zero-Inflated Power Series Dis-
tributions,” Communications in Statistics: Theory and
Methods, Vol. 40, No. 14, 2011, pp. 2539-2553.
doi:10.1080/03610926.2010.489172
[10] A. G. Arbous and J. E. Kerrich, “Accident Statistics and
the Concept of Accident Proneness,” Biometrics, Vol. 7,
No. 1951, pp. 340-342. doi:10.2307/3001656