Research Article
Volume 5 Issue 1  2017
On Discrete PoissonShanker Distribution and Its Applications
Rama Shanker^{1}*, Hagos Fesshaye^{2}, Ravi Shanker^{3}, Tekie Asehun Leonida^{4} and Simon Sium^{1}
^{1}Department of Statistics, Eritrea Institute of Technology, Eritrea
^{2}Department of Economics, College of Business and Economics, Eritrea
^{3}Department of Mathematics, GLA College NP University, India
^{4}Department of Applied Mathematics, University of Twente, Netherlands
Received: December 13, 2016  Published: January 18, 2017
*Corresponding author:
Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation:
Shanker R, Fesshaye H, Shanker R, Leonida TA, Sium S (2017) On Discrete PoissonShanker Distribution and Its Applications. Biom Biostat Int
J 5(1): 00121. DOI:
10.15406/bbij.2017.05.00121
Abstract
A simple method for obtaining moments of PoissonShanker distribution (PSD) introduced by Shanker [1] has been proposed. The first four moments about origin and the variance have been obtained. The goodness of fit and the applications of the PSD have been discussed with count data from ecology, genetics and thunderstorms and the fit is compared with one parameter Poisson distribution (PD) and PoissonLindley distribution (PLD) introduced by Sankaran [2].
Keywords: Shanker distribution; PoissonShanker distribution; PoissonLindley distribution; Moments; Estimation of parameter; Applications
Introduction
The PoissonShanker distribution (PSD) defined by its probability mass function
$P\left(X=x\right)=\frac{{\theta}^{2}}{{\theta}^{2}+1}\frac{x+\left({\theta}^{2}+\theta +1\right)}{{\left(\theta +1\right)}^{x+2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,\mathrm{...},\theta >0$
(1.1)
has been introduced by Shanker [1] for modeling count datasets. Shanker [1] has shown that PSD is a Poisson mixture of Shanker distribution introduced by Shanker [3] when the parameter
$\lambda $
of the Poisson distribution follows Shanker distribution of Shanker [3] having probability density function
$f\left(\lambda ;\theta \right)=\frac{{\theta}^{2}}{{\theta}^{2}+1}\left(\theta +\lambda \right){e}^{\theta \text{\hspace{0.17em}}\lambda}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};\lambda >0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0$
(1.2)
We have
$P\left(X=x\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{{e}^{\lambda}{\lambda}^{x}}{x!}}\cdot \frac{{\theta}^{2}}{{\theta}^{2}+1}\left(\theta +\lambda \right){e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
(1.3)
$=\frac{{\theta}^{2}}{\left({\theta}^{2}+1\right)\text{\hspace{0.17em}}x!}{\displaystyle \underset{0}{\overset{\infty}{\int}}{\lambda}^{x}}\left(\theta +\lambda \right){e}^{\left(\theta +1\right)\text{\hspace{0.17em}}\lambda}d\lambda $
$=\frac{{\theta}^{2}}{{\theta}^{2}+1}\frac{x+\left({\theta}^{2}+\theta +1\right)}{{\left(\theta +1\right)}^{x+2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,\mathrm{...},\theta >0$
. (1.4)
Which is the PoissonShanker distribution (PSD), as given in (1.1).
Shanker [3] has shown that the Shanker distribution (1.2) is a two component mixture of an exponential (
$\theta $
) distribution, a gamma (2,
$\theta $
) distribution with their mixing proportions
$\frac{{\theta}^{2}}{{\theta}^{2}+1}$
and
$\frac{1}{{\theta}^{2}+1}$
respectively. Shanker [3] has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stressstrength reliability , amongst others along with estimation of parameter and applications. Shanker & Hagos [4] have detailed study on modeling lifetime data using one parameter Akash distribution introduced by Shanker [5], Shanker distribution of Shanker [3], Lindley [6] distribution and exponential distribution.
The probability mass function of PoissonLindley distribution (PLD) given by
$P\left(X=x\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\theta}^{2}\text{\hspace{0.17em}}\left(x+\text{\hspace{0.17em}}\theta +\text{\hspace{0.17em}}2\right)}{{\left(\theta +\text{\hspace{0.17em}}1\right)}^{x+\text{\hspace{0.17em}}3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};$
x = 0, 1, 2,…,
$\theta >0$
. (1.5)
has been introduced by Sankaran [2] to model count data. The distribution arises from the Poisson distribution when its parameter
$\lambda $
follows Lindley [6] distribution with its probability density function
$f\text{\hspace{0.17em}}\left(\lambda ,\text{\hspace{0.17em}}\theta \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\theta}^{2}}{\theta +\text{\hspace{0.17em}}1}\text{\hspace{0.17em}}\left(1+\text{\hspace{0.17em}}\lambda \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}^{\theta \text{\hspace{0.17em}}\lambda}$
;
$x>0\text{\hspace{0.17em}},\theta >0$
(1.6)
Shanker et al. [7] have critical study on modeling of lifetime data using exponential and Lindley [6] distributions and observed that in some data sets Lindley distribution gives better fit than exponential distribution while in some data sets exponential distribution gives better fit than Lindley distribution. Shanker & Hagos [8] have detailed study on PoissonLindley distribution and its applications to model count data from biological sciences.
In this paper a simple method of finding moments of PoissonShanker distribution (PSD) introduced by Shanker [1] has been suggested and hence the first four moments about origin and the variance have been presented. It seems that not much work has been done on the applications of PSD so far. The PSD has been fitted to the some data sets relating to ecology, genetics and thunderstorms and the fit is compared with Poisson distribution (PD), and the PoissonLindley distribution (PLD). The goodness of fit of PSD shows satisfactory fit in majority of data sets.
Moments
Using (1.3) the
$r$
^{th} moment about origin of PSD (1.1) can be obtained as
${\mu}_{r}{}^{\prime}=E\left[E\left({X}^{r}\lambda \right)\right]=\frac{{\theta}^{2}}{{\theta}^{2}+1}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left[{\displaystyle \sum _{x=0}^{\infty}{x}^{r}\frac{{e}^{\lambda}{\lambda}^{x}}{x!}}\right]}\left(\theta +\lambda \right){e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
(2.1)
It is clear that the expression under the bracket in (2.1) is the
$r$
^{th} moment about origin of the Poisson distribution. Taking
$r=1$
in (2.1) and using the first moment about origin of the Poisson distribution, the first moment about origin of the PSD (1.1) can be obtained as
${\mu}_{1}{}^{\prime}=\frac{{\theta}^{2}}{{\theta}^{2}+1}{\displaystyle \underset{0}{\overset{\infty}{\int}}\lambda \left(\theta +\lambda \right)\text{\hspace{0.17em}}}{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda =\frac{{\theta}^{2}+2}{\theta \left({\theta}^{2}+1\right)}$
Again taking
$r=2$
in (2.1) and using the second moment about origin of the Poisson distribution, the second moment about origin of the PSD (1.1) can be obtained as
${\mu}_{2}{}^{\prime}=\frac{{\theta}^{2}}{{\theta}^{2}+1}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left({\lambda}^{2}+\lambda \right)\left(\theta +\lambda \right)\text{\hspace{0.17em}}}{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda =\frac{{\theta}^{3}+2{\theta}^{2}+2\theta +6}{{\theta}^{2}\left({\theta}^{2}+1\right)}$
Similarly, taking
$r=3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4$
in (2.1) and using the third and fourth moments about origin of the Poisson distribution, the third and the fourth moments about origin of the PSD (1.1) are obtained as
${\mu}_{3}{}^{\prime}=\frac{{\theta}^{4}+6{\theta}^{3}+8{\theta}^{2}+18\theta +24}{{\theta}^{3}\left({\theta}^{2}+1\right)}$
${\mu}_{4}{}^{\prime}=\frac{{\theta}^{5}+14{\theta}^{4}+38{\theta}^{3}+66{\theta}^{2}+144\theta +120}{{\theta}^{4}\left({\theta}^{2}+1\right)}$
The variance of PoissonShanker distribution can thus be obtained as
${\mu}_{2}={\sigma}^{2}=\frac{{\theta}^{5}+{\theta}^{4}+3{\theta}^{3}+4{\theta}^{2}+2\theta +2}{{\theta}^{2}{\left({\theta}^{2}+1\right)}^{2}}$
Estimation of Parameter
Maximum likelihood estimate (MLE) of the parameter: Suppose
$\left({x}_{1},{x}_{2},\mathrm{...},{x}_{n}\right)$
is a random sample of size
$n$
from the PSD (1.1) and suppose
${f}_{x}$
be the observed frequency in the sample corresponding to
$X=x\text{\hspace{0.17em}}\text{\hspace{0.17em}}(x=1,2,3,\mathrm{...},k)$
such that
$\sum _{x=1}^{k}{f}_{x}}=n$
, where
$k$
is the largest observed value having nonzero frequency. The likelihood function
$L$
of the PSD (1.1) is given by
$L={\left(\frac{{\theta}^{2}}{{\theta}^{2}+1}\right)}^{n}\frac{1}{{\left(\theta +1\right)}^{{\displaystyle \sum _{x=1}^{k}{f}_{x}\left(x+2\right)}}}{\displaystyle \prod _{x=1}^{k}{\left[x+\left({\theta}^{2}+\theta +1\right)\right]}^{{f}_{x}}}$
The log likelihood function is thus obtained as
$\mathrm{log}L=n\mathrm{log}\left(\frac{{\theta}^{2}}{{\theta}^{2}+1}\right){\displaystyle \sum _{x=1}^{k}{f}_{x}\left(x+2\right)}\mathrm{log}\left(\theta +1\right)+{\displaystyle \sum _{x=1}^{k}{f}_{x}\mathrm{log}\left[x+\left({\theta}^{2}+\theta +1\right)\right]}$
The first derivative of the log likelihood function is given by
$\frac{d\mathrm{log}L}{d\theta}=\frac{2n}{\theta \left({\theta}^{2}+1\right)}\frac{n\left(\overline{x}+2\right)}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{\left(2\theta +1\right){f}_{x}}{x+\left({\theta}^{2}+\theta +1\right)}}$
where
$\overline{x}$
is the sample mean.
The maximum likelihood estimate (MLE),
$\widehat{\theta}$
of
$\theta $
of PSD (1.1) is the solution of the following nonlinear equation
$\frac{2n}{\theta \left({\theta}^{2}+1\right)}\frac{n\left(\overline{x}+2\right)}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{\left(2\theta +1\right){f}_{x}}{x+\left({\theta}^{2}+\theta +1\right)}}=0$
This nonlinear equation can be solved by any numerical iteration methods such as NewtonRaphson method, Bisection method, RegulaFalsi method etc. In this paper, NewtonRaphson method has been used for estimating the parameter.
Shanker [1] has showed that the MLE of
$\theta $
of PSD (1.1) is consistent and asymptotically normal.
Method of moment estimate (MOME) of the parameter: Equating the population mean to the corresponding sample mean, the MOME
$\tilde{\theta}$
of
$\theta $
of PSD (1.1) is the solution of the following cubic equation
$\overline{x}{\theta}^{3}{\theta}^{2}+\overline{x}\theta 2=0$
where
$\overline{x}$
is the sample mean.
Goodness of Fit and Applications
Since the condition for the applications for Poisson distribution is the independence of events and equality of mean and variance, this condition is rarely satisfied completely in biological and medical science due to the fact that the occurrences of successive events are dependent. Further, the negative binomial distribution is a possible alternative to the Poisson distribution when successive events are possibly dependent, (see Johnson et al. [9]), but for fitting negative binomial distribution (NBD) to the count data, mean should be less than the variance (overdispersion). In biological and medical sciences, these conditions are not fully satisfied. Generally, the count data in biological science and medical science are either overdispersed or underdispersed. The main reason for selecting PLD and PSD to fit data from biological science and thunderstorms are that these two distributions are always overdispersed and PSD has some flexibility over PLD.
Count data from ecology and biological sciences
In this section we fit Poisson distribution (PD), Poisson Lindley distribution (PLD) and PoissonShanker distribution (PSD) to many count data from ecology and biological sciences using maximum likelihood estimate. The data were on haemocytometer yeast cell counts per square, on European red mites on apple leaves and European corn borers per plant. Recall that Shanker & Hagos [7] have fitted PoissonLindley distribution(PLD) to the same data sets.
It is obvious from above tables that in Table 1, PD gives better fit than PLD and PSD; in Table 2, PSD gives better fit than PD and PLD while in Table 3, PLD gives better fit than PD and PSD.
Number of Yeast Cells per Square 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
213 
202.1 
234.0 
233.2 
1 
128 
138.0 
99.4 
99.6 
2 
37 
47.1 
40.5 
41.0 
3 
18 
$\begin{array}{l}10.7\\ 1.8\\ 0.2\\ 0.1\end{array}\}$

$\begin{array}{l}16.0\\ 6.2\\ 2.4\\ 1.5\end{array}\}$

$\begin{array}{l}16.3\\ 6.7\\ 2.3\\ 0.9\end{array}\}$

4 
3 
5 
1 
6 
0 
Total 
400 
400.0 
400.0 
400.0 
ML Estimate 

$\widehat{\theta}=0.6825$

$\widehat{\theta}=1.950236$

$\widehat{\theta}=1.795126$

${\chi}^{2}$


10.08 
11.04 
12.25 
d.f. 

2 
2 
2 
pvalue 

0.0065 
0.0040 
0.0023 
Table 1: Observed and expected number of Haemocytometer yeast cell counts per square observed by Gosset [10].
Number mites per Leaf 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
38 
25.3 
35.8 
36.0 
1 
17 
29.1 
20.7 
20.6 
2 
10 
16.7 
11.4 
11.2 
3 
9 
$\begin{array}{l}6.4\\ 1.8\\ 0.4\\ 0.2\\ 0.1\end{array}\}$

6.0 
6.0 
4 
3 
$\begin{array}{l}3.1\\ 1.6\\ 0.8\\ 0.6\end{array}\}$

$\begin{array}{l}3.1\\ 1.6\\ 0.8\\ 0.7\end{array}\}$

5 
2 
6 
1 
7+ 
0 
Total 
80 
80.0 
80.0 
80.0 
ML Estimate 

$\widehat{\theta}=1.15$

$\widehat{\theta}=1.255891$

$\widehat{\theta}=1.219731$

${\chi}^{2}$


18.27 
2.47 
2.37 
d.f. 

2 
3 
3 
pvalue 

0.0001 
0.4807 
0.4992 
Table 2: Observed and expected number of red mites on Apple leaves, available in Fisher et al. [11].
Number of bores per Plant 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
188 
169.4 
194.0 
195.0 
1 
83 
109.8 
79.5 
78.4 
2 
36 
35.6 
31.3 
31.0 
3 
14 
$\begin{array}{l}7.8\\ 1.2\\ 0.2\end{array}\}$

$\begin{array}{l}12.0\\ 4.5\\ 2.7\end{array}\}$

$\begin{array}{l}12.1\\ 4.6\\ 2.9\end{array}\}$

4 
2 
5 
1 
Total 
324 
324.0 
324.0 
324.0 
ML Estimate 

$\widehat{\theta}=0.648148$

$\widehat{\theta}=2.043252$

$\widehat{\theta}=1.879553$

${\chi}^{2}$


15.19 
1.29 
1.67 
d.f. 

2 
2 
2 
pvalue 

0.0005 
0.5247 
0.4338 
Table 3: Observed and expected number of European corn borer of Mc Guire et al. [12].
Number of Aberrations 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
268 
231.3 
257.0 
258.3 
1 
87 
126.7 
93.4 
92.1 
2 
26 
34.7 
32.8 
32.4 
3 
9 
$\begin{array}{l}6.3\\ 0.8\\ 0.1\\ 0.1\\ 0.1\end{array}\}$

11.2 
11.3 
4 
4 
$\begin{array}{l}3.8\\ 1.2\\ 0.4\\ 0.2\end{array}\}$

$\begin{array}{l}3.9\\ 1.3\\ 0.5\\ 1.5\end{array}\}$

5 
2 
6 
1 
7+ 
3 
Total 
400 
400.0 
400.0 
400.0 
ML Estimate 

$\widehat{\theta}=0.5475$

$\widehat{\theta}=2.380442$

$\widehat{\theta}=2.162674$

${\chi}^{2}$


38.21 
6.21 
3.45 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.1018 
0.3273 
Table 4: Distribution of number of Chromatid aberrations (0.2 g chinon 1, 24 hours).
Count data from genetics
In this section we fit PSD, PLD and PD using maximum likelihood estimate to count data relating to genetics. Recall that Shanker & Hagos [8] have fitted PoissonLindley distribution to the same data sets. The data set in Table 4 is available in Loeschke & Kohler [13], and Janardan & Schaeffer [14]. The data sets in Tables 57 are available in Catcheside et al. [15,16].
Class/Exposure (
$\mu gkg$
) 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
413 
374.0 
405.7 
407.1 
1 
124 
177.4 
133.6 
131.9 
2 
42 
42.1 
42.6 
42.3 
3 
15 
$\begin{array}{l}6.6\\ 0.8\\ 0.1\\ 0.0\end{array}\}$

13.3 
13.5 
4 
5 
$\begin{array}{l}4.1\\ 1.2\\ 0.5\end{array}\}$

$\begin{array}{l}4.3\\ 1.3\\ 0.6\end{array}\}$

5 
0 
6 
2 
Total 
601 
601.0 
601.0 
601.0 
ML Estimate 

$\widehat{\theta}=0.47421$

$\widehat{\theta}=2.685373$

$\widehat{\theta}=2.419447$

${\chi}^{2}$


48.17 
1.34 
0.82 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.7196 
0.8446 
Table 5: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure60
$\mu gkg$
.
Class/Exposure (
$\mu gkg$
) 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
200 
172.5 
191.8 
192.7 
1 
57 
95.4 
70.3 
69.4 
2 
30 
26.4 
24.9 
24.6 
3 
7 
$\begin{array}{l}4.9\\ 0.7\\ 0.1\\ 0.0\end{array}\}$

$\begin{array}{l}8.6\\ 2.9\\ 1.0\\ 0.5\end{array}\}$

$\begin{array}{l}8.7\\ 3.0\\ 1.0\\ 0.6\end{array}\}$

4 
4 
5 
0 
6 
2 
Total 
300 
300.0 
300.0 
300.0 
ML Estimate 

$\widehat{\theta}=0.55333$

$\widehat{\theta}=2.353339$

$\widehat{\theta}=2.138048$

${\chi}^{2}$


29.68 
3.91 
3.66 
d.f. 

2 
2 
2 
pvalue 

0.0000 
0.1415 
0.1604 
Table 6: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure70
$\mu gkg$
.
Class/Exposure (
$\mu gkg$
) 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
155 
127.8 
158.3 
159.3 
1 
83 
109.0 
77.2 
76.3 
2 
33 
46.5 
35.9 
35.4 
3 
14 
$\begin{array}{l}13.2\\ 2.8\\ 0.5\\ 0.2\end{array}\}$

16.1 
16.1 
4 
11 
$\begin{array}{l}7.1\\ 3.1\\ 2.3\end{array}\}$

$\begin{array}{l}7.2\\ 3.2\\ 2.5\end{array}\}$

5 
3 
6 
1 
Total 
300 
300.0 
300.0 
300.0 
ML Estimate 

$\widehat{\theta}=0.853333$

$\widehat{\theta}=1.617611$

$\widehat{\theta}=1.520805$

${\chi}^{2}$


24.97 
1.51 
1.48 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.6799 
0.6868 
Table 7: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure 90
$\mu gkg$
.
It is obvious from the fitting of PSD, PLD, and PD that both PSD and PLD gives much satisfactory fit than PD. Further, PSD gives much closer fit than both PLD and PD in almost all data sets.
Count data from thunderstorms
In this section, we fit PSD, PLD and PD to count data from thunderstorms available in Falls et al. [17].
It is obvious from the fitting of PSD, PLD and PD to thunderstorms data that PLD gives better fit than both PSD and PD in Table 8, 9 and 11 while PSD gives better fit than both PLD and PD in Table 10.
No. of Thunderstorms 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
187 
155.6 
185.3 
186.4 
1 
77 
117.0 
83.5 
82.3 
2 
40 
43.9 
35.9 
35.5 
3 
17 
$\begin{array}{l}11.0\\ 2.1\\ 0.3\\ 0.1\end{array}\}$

15.0 
15.0 
4 
6 
$\begin{array}{l}6.1\\ 2.5\\ 1.7\end{array}\}$

$\begin{array}{l}6.3\\ 2.6\\ 1.9\end{array}\}$

5 
2 
6 
1 
Total 
330 
330.0 
330.0 
330.0 
ML Estimate 

$\widehat{\theta}=0.751515$

$\widehat{\theta}=1.804268$

$\widehat{\theta}=1.679053$

${\chi}^{2}$


31.93 
1.43 
1.48 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.6985 
0.6869 
Table 8: Observed and expected number of days that experienced X thunderstorms events at Cape Kennedy, Florida for the 11year period of record for the month of June, January 1957 to December 1967, Falls et al. [17].
No. of Thunderstorms 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
177 
142.3 
177.7 
178.7 
1 
80 
124.4 
88.0 
86.9 
2 
47 
54.3 
41.5 
41.0 
3 
26 
$\begin{array}{l}15.8\\ 3.5\\ 0.7\end{array}\}$

18.9 
18.9 
4 
9 
$\begin{array}{l}8.4\\ 6.5\end{array}\}$

$\begin{array}{l}8.6\\ 6.9\end{array}\}$

5 
2 
Total 
341 
341.0 
341.0 
341.0 
ML Estimate 

$\widehat{\theta}=0.873900$

$\widehat{\theta}=1.583536$

$\widehat{\theta}=1.497274$

${\chi}^{2}$


39.74 
5.15 
5.41 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.1611 
0.1441 
Table 9: Observed and expected number of days that experienced X thunderstorms events at Cape Kennedy, Florida for the 11year period of record for the month of July, January 1957 to December 1967, Falls et al. [17].
No. of Thunderstorms 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
185 
151.8 
184.8 
186.0 
1 
89 
122.9 
87.2 
86.1 
2 
30 
49.7 
39.3 
38.8 
3 
24 
$\begin{array}{l}13.4\\ 2.7\\ 0.5\end{array}\}$

17.1 
17.1 
4 
10 
$\begin{array}{l}7.3\\ 5.3\end{array}\}$

$\begin{array}{l}7.4\\ 5.6\end{array}\}$

5 
3 
Total 
341 
341.0 
341.0 
341.0 
ML estimate 

$\widehat{\theta}=0.809384$

$\widehat{\theta}=1.693425$

$\widehat{\theta}=1.586731$

${\chi}^{2}$


49.49 
5.03 
4.87 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.1696 
0.1816 
Table 10: Observed and expected number of days that experienced X thunderstorms events at Cape Kennedy, Florida for the 11year period of record for the month of August, January 1957 to December 1967, Falls et al. [17].
No. of Thunderstorms 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
549 
547.5 
547.5 
550.8 
1 
246 
364.8 
259.0 
255.7 
2 
117 
148.2 
116.9 
115.5 
3 
67 
40.1 
51.2 
51.1 
4 
25 
$\begin{array}{l}8.1\\ 1.3\\ 0.3\end{array}\}$

21.9 
22.3 
5 
7 
$\begin{array}{l}9.2\\ 6.3\end{array}\}$

$\begin{array}{l}9.6\\ 7.0\end{array}\}$

6 
1 
Total 
1012 
1012.0 
1012.0 
1012.0 
ML Estimate 

$\widehat{\theta}=0.812253$

$\widehat{\theta}=1.688990$

$\widehat{\theta}=1.582475$

${\chi}^{2}$


141.42 
9.60 
10.09 
d.f. 

3 
4 
4 
pvalue 

0.0000 
0.0477 
0.0389 
Table 11: Observed and expected number of days that experienced X thunderstorms events at Cape Kennedy, Florida for the 11year period of record for the summer, January 1957 to December 1967, Falls et al. [17].
Concluding Remarks
In the present paper, a simple and interesting method for finding moments of PoissonShanker distribution (PSD) has been suggested and thus the first four moments about origin and the variance have been obtained. The goodness of fit of PSD has been discussed with several data from ecology, genetics and thunderstorms and the fit has been compared with Poisson distribution (PD) and PoissonLindley distribution (PLD).
References
 Shanker R (2016) The discrete PoissonShanker distribution. Jacobs Journal of Biostatistics 1(1): 17.
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