41 Chapter 3 Inference for Zero-Inflated Poisson Distribution 3.0 Introduction In chapter two, we have discussed inference for θ of Zero-Inflated Power Series Distribution. Zero-Inflated Poisson Distribution is a particular case of Zero-Inflated Power Series Distribution. In this chapter, we provide the inference for Zero-Inflated Poisson Distribution and Zero-Inflated Truncated Poisson Distribution. In the literature, numbers of researchers have worked on zero-inflated Poisson distribution. Yip (1988) has described an inflated Poisson distribution dealing with the number of insects per leaf. Lambert (1992) considered zero-inflated Poisson regression model. Van Den Broek (1995) has discussed score test for testing Poisson distribution against ZIP distribution. Xie et al. (2001) have reported use of ZIP distribution in statistical process control and studied performance of various tests for testing Poisson distribution against zero-inflated Poisson alternative. Gupta et al. (2004) have discussed score test for zero-inflated generalized Poisson regression model. Thas et al. (2005) have discussed smooth tests for the zero-
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41
Chapter 3
Inference for Zero-Inflated Poisson Distribution
3.0 Introduction
In chapter two, we have discussed inference for θ of Zero-Inflated
Power Series Distribution. Zero-Inflated Poisson Distribution is a particular
case of Zero-Inflated Power Series Distribution. In this chapter, we provide
the inference for Zero-Inflated Poisson Distribution and Zero-Inflated
Truncated Poisson Distribution. In the literature, numbers of researchers have
worked on zero-inflated Poisson distribution. Yip (1988) has described an
inflated Poisson distribution dealing with the number of insects per leaf.
Lambert (1992) considered zero-inflated Poisson regression model. Van Den
Broek (1995) has discussed score test for testing Poisson distribution against
ZIP distribution. Xie et al. (2001) have reported use of ZIP distribution in
statistical process control and studied performance of various tests for testing
Poisson distribution against zero-inflated Poisson alternative. Gupta et al.
(2004) have discussed score test for zero-inflated generalized Poisson
regression model. Thas et al. (2005) have discussed smooth tests for the zero-
42
inflated Poisson distribution. Castillo et al. (2005) have studied overdispersed
and underdispersed Poisson distributions.
ZIPD contains two parameters. The first parameter )(π indicates
inflation of zero and the other parameter )(θ is that of Poisson distribution. In
this chapter, we focus on the inference of parameter θ of ZIPD. However it
appears that no test has been reported for testing parameter θ of Poisson
distribution. We provide maximum likelihood estimators, Fisher information
matrix and moment estimator of the parameters. The three asymptotic tests for
testing the parameter of Poisson distribution based on full likelihood,
conditional likelihood and moment estimator are provided. The performance
of these three tests has been studied for ZIPD. Asymptotic confidence
intervals for the parameter are also provided.
The rest of the chapter is organized as follows. In section 3.1, we
report maximum likelihood estimators of both the parameters of ZIPSD and
corresponding asymptotic variances using full likelihood and conditional
likelihood approach. In section 3.2, we provide three asymptotic tests for
testing the parameter of Poisson distribution. In section 3.3, the performance
of the three tests has been studied by simulation. We have developed C -
program for the same (Appendix 1). Based on study it is observed that an
asymptotic test based on full likelihood estimator and the one based on
conditional likelihood estimator have nearly similar performance. In section
3.4, the asymptotic confidence intervals based on three estimation approaches
are given. Section 3.5 is devoted to ZITPD. The ZITPD is a member of
ZIPSD. All the theory is analogues to ZIPD. Section 3.6 devoted to three tests
for testing the parameters of ZITPD.
3.1 Zero-Inflated Poisson Distribution
Full Likelihood Function Approach
Let nXXX ...,,, 21 be a random sample observed from ZIP distribution, then
the likelihood function is given by
43
( ) 0,!
1);,(
1
1
>
+−=
−−
=
−∏ πθθπ
πππθθ
θi
iia
xan
i x
eexL …(3.1.1)
The corresponding log likelihood function is given by
=);,(log xL πθ
∑∑ ∑∑== ==
− −+−++−=n
i
ii
n
i
n
i
ii
n
i
ii xaxaaaen11 11
0 !log)log(log)1log( θθπππ θ
…(3.1.2)
Suppose θ̂ and π̂ are mles of θ and π respectively, then )1(
ˆˆ
0
θπ
−−
−=
en
nn and
the corresponding mle of θ is the solution to equation ( )θθ
ˆ1
ˆ
−−=
ex
In the following we find the elements of Fisher information matrix.
We have, ( )
( ) ππππ θ
θ ∑=
−
−
++−
+−=
∂∂
n
i
ia
e
enL 10
1
1log,
and 2
1
2
20
2
2
)1(
)1(log
ππππ θ
θ ∑=
−
−
−+−
+−−=
∂
∂
n
i
ia
e
enL.
Therefore,
2
1
02
2
2
2
)()1(
)1(log
ππππ θ
θ
++−
+−=
∂
∂−
∑=
−
−
n
i
iaE
nEe
eLE ,
22
2 )1(
)1(
)1()1(
ππ
ππππ θ
θ
θθ −
−
−− −+
+−
+−+−=
en
e
ene.
We note that, ( ) )1(0
θππ −+−= ennE and )1(1
θπ −
=
−=
∑ enaE
n
i
i .
Hence,
ππππ
θ
θ
θ )1(
)1(
)1(log 2
2
2 −
−
− −+
+−
+−=
∂
∂−
en
e
neLE ,
44
)1(
)1(θ
θ
πππ −
−
+−
−=
e
en. …(3.1.3)
Now
θπππ
θ θ
θ ∑∑ =
=−
−
+−+−
−=∂∂
n
i
iin
i
i
xa
ae
enL 1
1
0
)1(
)(log
2
1
2
0
2
2
)1(
)1(log
θππππ
θ θ
θ ∑=
−
−
−+−
−−=
∂
∂
n
i
ii xa
e
enL
Therefore,
+
+−
−=
∂
∂−
−
−
θπππ
πθ θ
θ 1
)1(
)1(log2
2
e
en
LE …(3.1.4)
Further, ( )2
02
1
log
θ
θ
ππθπ −
−
+−−=
∂∂∂
e
enL
Hence
( )θθ
ππθπ −
−
+−=
∂∂∂
−e
enLE
1
log2
…(3.1.5)
Thus the elements of Fisher Information Matrix are
( )
( )
+−−
=−
−
θ
θ
eπππ
enI
1
111 ,
+
+−
−=
−
−
θeππ
eππnI
θ
θ1
1
)1(22 ,
θ
θ
ππ −
−
+−==
e
enII
12112
Therefore, asymptotic variance of θ̂ is given by
1
11
1222ˆ ),(
−
−=
I
IIVar θπ
θ and estimator of the same is )ˆ,ˆ(ˆ θπ
θVar .
…(3.1.6)
45
Conditional Likelihood Function Approach
The conditional likelihood function is given by
0,)1(!
);(1
>
−=∏=
−
−∗ θ
ex
θexθL
n
i
a
θi
xθ ii
…(3.1.7)
The corresponding log likelihood function is given by
∑∑−
=
−−
=
−−−−−−=00
1
00
1
!log)1log()()()log();(*lognn
i
i
nn
i
i xennnnxxL θθθθ
−+−−=
∂∂
−
−−
=∑ θ
θ
θθ e
ennx
L nn
i
i1
1)(1*log
0
1
0
Equating θ∂
∂ *log L equal to zero we get the mle of θ .
The corresponding mle θ~
is the solution to equation,
θ
e
θx ~
1
~
−−= …(3.1.8)
Now consider,
2
0
122
2
)1(
)(1*log 0
θ
θ
θθ −
−−
= −
−+−=
∂
∂∑
e
ennx
L nn
i
i .
Therefore,
2
0
122
2
)1(
)(1*log 0
θ
θ
θθ −
−−
= −
−−
=
∂
∂− ∑
e
ennxE
LE
nn
i
i ,
2
00
2 )1(
)(
)1(
)(1θ
θ
θ
θθ −
−
− −
−−
−
−=
e
enn
e
nn,
−−
−
−=
−
−
− )1(
1
)1(
)( 0
θ
θ
θ θ e
e
e
nn.
The asymptotic variance of θ~
is
1
0~
)1(
1
)1(
)()(
−
−
−
−
−−
−
−=
θ
θ
θθe
e
θe
nnθAV …(3.1.9)
46
Moment Estimator of ZIP Distribution
For the zero-inflated Poisson distribution the mean and variance are
given by, θπ== XXE )( and ( ))1(1)( 2 πθπθ −+== SXVar …(3.1.10)
Therefore,
θπ=)(XE and ( )
),()1(1
)( 2 θπσπθπθ=
−+=
nXVar say …(3.1.11)
Let θπ== XXE )( …(3.1.12)
and ( ))1(12 πθπθ −+=S
( )πθθπθ −+= 1
( )XX −+= θ1
2XXX −+= θ
22XXSX +−=θ
X
XXS 22
ˆ +−=θ
)1(ˆ2
XX
S−−=θ
Now, θ
πˆ
ˆX=
Hence, )1(
ˆ2
2
XXS
X
−−=π
22
2
XXS
X
+−= …(3.1.13)
Solving Eq. (3.1.12) and Eq. (3.1.13) we get the moment estimators of π and
θ .
3.2 Tests for the Parameter θ of ZIP Distribution
Let us consider the problem of testing 00 : θθ =H . We assume that π is
unknown.
a) Test Based On θ̂
The test statistic for testing 00 : θθ =H vs 01 : θθ ≠H , is given by
47
),ˆ(
ˆ
00ˆ
07
θπ
θθ
θAV
Z−
= , …(3.2.1)
where )( 0ˆ θθAV is an estimate of asymptotic variance of θ̂ . The test 7ψ
rejects 0H , if 2/17 α−> zZ
b) Test Based On θ~
The test statistic here is: )(
~
0~
08
θ
θθ
θAVZ
−= , …(3.2.2)
where )( 0~ θθAV is as defined in Eq. (3.1.9). The test 8ψ rejects 0H if
2/18 α−> zZ .
c) Test Based On Sample Mean
The test statistic is
),ˆ(ˆ
ˆ
00
2
0
0
0
9
θππ
θπ
XAV
Xn
Z−
−
= , …(3.2.3)
where, 0
0ˆθ
πX= ,
Power of the test is given by
),(9θπβψ ( )∑
=
=Φ+Φ−=n
k
kk knPAB0
0 )()ˆ()ˆ(1
where ( )
),(
),ˆ(ˆˆˆ
0
00
2
02/100
θπ
θπθππθπ α
X
X
kAV
AVzB
−+=
−−
,
( )
),(
),ˆ(ˆˆˆ
0
0
2
02/100
θπ
θπθππθπ α
X
X
kAV
AVzA
−−=
−−
and
( ) knkn
k PPknP −−== )1()( 000 , with ( )θππ −+−= eP 10
48
In the following, we report performance of the three tests developed in
the section (3.2), which is based on simulation experiments.
3.3 Simulation Study
A simulation study is carried out to investigate the power of the three
tests proposed in the section 3.2. We generate 25000 samples of sizes 50 and
100 for different values of θ and π . Based on the generated sample, the test
statistics were calculated. Percentage of times the test statistic exceeds 2/1 α−z
is computed. It is infact an estimate of power of the respective test. ‘C’
programs are developed to find power of the test (Appendix 1). The results for
the case of =0θ 2 and 5 and =π 0.3, 0.4, 0.5, 0.6, 0.7 are presented in the
Table 3.3.1 and Table 3.3.2.
49
Table 3.3.1: Power (in %) of the test 7ψ , 8ψ and 9ψ for 20 =θ