Research Article
Volume 3 Issue 6  2016
On PoissonAkash Distribution and its Applications
Rama Shanker^{1}*, Hagos Fesshaye^{2} and Teklay Tesfazghi^{3}
^{1}Department of Statistics, Eritrea Institute of Technology, Eritrea
^{2}Department of Economics, College of Business and Economics, Eritrea
^{3}Department of Computer Engineering, Eritrea Institute of Technology, Eritrea
Received:April 25, 2016  Published: May 11, 2016
*Corresponding author:
Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation:
Shanker R, Fesshaye H, Tesfazghi T (2016) On PoissonAkash Distribution and its Applications. Biom Biostat Int J 3(6): 00075. DOI:
10.15406/bbij.2016.03.00075
Abstract
A simple and interesting method for finding moments of ‘PoissonAkash distribution (PAD)’ of Shanker [1], a Poisson mixture of Akash distribution introduced by Shanker [2] has been suggested. The first two moments about origin and the variance of PAD has been obtained and presented. The applications and the goodness of fit of PAD has been discussed using datasets relating to ecology genetics, and thunderstorms and the fit has been compared with Poisson and PoissonLindley distribution, a Poisson mixture of Lindley [3] distribution, introduced by Sankaran [4] and the goodness of fit of PAD shows satisfactory fit in most of datasets.
Keywords: Akash distribution; PoissonAkash distribution; Lindley distribution; PoissonLindley distribution; Compounding; Moments; Estimation of parameter; Goodness of fit
Introduction
The probability mass function of PoissonAkash distribution (PAD) having parameter
$\theta $
given by
$P\left(X=x\right)=\frac{{\theta}^{3}}{{\theta}^{2}+2}\cdot \frac{{x}^{2}+3x+\left({\theta}^{2}+2\theta +3\right)}{{\left(\theta +1\right)}^{x+3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,\mathrm{...},\theta >0$
(1.1)
has been introduced by Shanker [1] for modeling various count datasets. The PAD arises from Poisson distribution when its parameter
$\lambda $
follows one parameter Akash distribution introduced by Shanker [2] having probability density function
$f\left(\lambda ,\text{\hspace{0.17em}}\theta \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\theta}^{3}}{{\theta}^{2}+\text{\hspace{0.17em}}2}\text{\hspace{0.17em}}\left(\text{\hspace{0.17em}}1+\text{\hspace{0.17em}}{\lambda}^{2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}^{\theta \text{\hspace{0.17em}}\lambda}\text{\hspace{0.17em}}\text{\hspace{0.17em}};\lambda >0,\text{\hspace{0.17em}}\theta >0$
(1.2)
We have
$P\left(X=x\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{{e}^{\lambda}{\lambda}^{x}}{x!}}\cdot \frac{{\theta}^{3}}{{\theta}^{2}+2}\left(1+{\lambda}^{2}\right){e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
(1.3)
$=\frac{{\theta}^{3}}{\left({\theta}^{2}+2\right)\text{\hspace{0.17em}}x!}{\displaystyle \underset{0}{\overset{\infty}{\int}}{e}^{\left(\theta +1\right)\text{\hspace{0.17em}}\lambda}\left[{\lambda}^{x}+{\lambda}^{x+2}\right]}\text{\hspace{0.17em}}d\lambda $
$=\frac{{\theta}^{3}}{{\theta}^{2}+2}\cdot \frac{{x}^{2}+3x+\left({\theta}^{2}+2\theta +3\right)}{{\left(\theta +1\right)}^{x+3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,\mathrm{...},\theta >0$
(1.4)
This is the probability mass function of PoissonAkash distribution (PAD)”.
It has been shown by Shanker [2] that the Akash distribution (1.2) is a two component mixture of an exponential (
$\theta $
) distribution, and a gamma (3,
$\theta $
) distribution with their mixing proportions
$\frac{{\theta}^{2}}{{\theta}^{2}+2}$
and
$\frac{2}{{\theta}^{2}+2}$
respectively. Shanker [2] has discussed its mathematical and statistical properties including its shape, 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 the estimation of parameter and applications for modeling lifetime data from engineering and biomedical science.
Sankaran [3] obtained PoissonLindley distribution (PLD) having probability mass function (p.m.f)
$P\left(X=x\right)=\frac{{\theta}^{2}\left(x+\theta +2\right)}{{\left(\theta +1\right)}^{x+3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,\mathrm{...},\theta >0$
(1.5)
by compounding Poisson distribution with Lindley distribution when the parameter
$\lambda $
of Poisson distribution follows Lindley distribution, introduced by Lindley [5] having probability density function (p.d.f)
$f\left(\lambda ,\theta \right)=\frac{{\theta}^{2}}{\theta +1}\left(1+\lambda \right)\text{\hspace{0.17em}}{e}^{\theta \text{\hspace{0.17em}}\text{}\lambda}\text{\hspace{0.17em}}\text{\hspace{0.17em}};\lambda >0,\text{\hspace{0.17em}}\theta >0$
(1.6)
In this paper a simple and interesting method for finding moments of PoissonAkash distribution (PAD) introduced by Shanker [5] has been suggested and hence the first two moments about origin and the variance has been presented. It seems that not much work has been done on the applications of PAD so far for count data arising in various fields of knowledge. The applications and goodness of fit of PAD have been discussed with various count data from ecology, genetics and thunderstorms and the goodness of fit of PAD has been compared with Poisson distribution and PoissonLindley distribution (PLD). The goodness of fit of PAD shows satisfactory fit in most of the datasets.
Moments of Pad
Using (1.3), the th moment about origin of PAD (1.1) can be obtained as
${\mu}_{r}{}^{\prime}=E\left[E\left({X}^{r}\lambda \right)\right]$
$=\frac{{\theta}^{3}}{{\theta}^{2}+2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left[{\displaystyle \sum _{x=0}^{\infty}{x}^{r}\frac{{e}^{\lambda}{\lambda}^{x}}{x!}}\right]}\left(1+{\lambda}^{2}\right){e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
It is obvious 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 PAD (1.1) can be obtained as
${\mu}_{1}{}^{\prime}=\frac{{\theta}^{3}}{{\theta}^{2}+2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\lambda \left(1+{\lambda}^{2}\right)}\text{\hspace{0.17em}}{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda =\frac{{\theta}^{2}+6}{\theta \left({\theta}^{2}+2\right)}$
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 PAD (1.1) can be obtained as
${\mu}_{2}{}^{\prime}=\frac{{\theta}^{3}}{{\theta}^{2}+2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left({\lambda}^{2}+\lambda \right)\left(1+{\lambda}^{2}\right)}\text{\hspace{0.17em}}{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda =\frac{{\theta}^{3}+2{\theta}^{2}+6\theta +24}{{\theta}^{2}\left({\theta}^{2}+2\right)}$
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 the fourth moments about origin of the Poisson distribution, the third and the fourth moments about origin of the PAD (1.1) can thus be obtained as
${\mu}_{3}{}^{\prime}=\frac{{\theta}^{4}+6{\theta}^{3}+12{\theta}^{2}+72\theta +120}{{\theta}^{3}\left({\theta}^{2}+2\right)}$
(2.4)
${\mu}_{4}{}^{\prime}=\frac{{\theta}^{5}+14{\theta}^{4}+42{\theta}^{3}+192{\theta}^{2}+720\theta +720}{{\theta}^{4}\left({\theta}^{2}+2\right)}$
(2.5)
The variance of the PAD (1.1) can thus be obtained as
${\mu}_{2}=\frac{{\theta}^{5}+{\theta}^{4}+8{\theta}^{3}+16{\theta}^{2}+12\theta +12}{{\theta}^{2}{\left({\theta}^{2}+2\right)}^{2}}$
(2.6)
It has been shown by Shanker [5] that PAD (1.1) has increasing hazard rate, unimodal and always overdispersed, and thus is a suitable model for count data which are overdispersed
Parameter Estimation of Pad
Maximum Likelihood Estimate (MLE) of the Parameter: Let
$\left({x}_{1},{x}_{2},\mathrm{...},{x}_{n}\right)$
be a random sample of size
$n$
from the PAD (1.1) and let
${f}_{x}$
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 PAD (1.1) can be given by
$L={\left(\frac{{\theta}^{3}}{{\theta}^{2}+2}\right)}^{n}\frac{1}{{\left(\theta +1\right)}^{{\displaystyle \sum _{x=1}^{k}{f}_{x}\left(x+3\right)}}}{\displaystyle \prod _{x=1}^{k}{\left[{x}^{3}+3x+\left({\theta}^{2}+2\theta +3\right)\right]}^{{f}_{x}}}$
The log likelihood function is thus obtained as
$\mathrm{log}L=n\mathrm{log}\left(\frac{{\theta}^{3}}{{\theta}^{2}+2}\right){\displaystyle \sum _{x=1}^{k}{f}_{x}\left(x+3\right)}\mathrm{log}\left(\theta +1\right)+{\displaystyle \sum _{x=1}^{k}{f}_{x}\mathrm{log}\left[{x}^{2}+x+\left({\theta}^{2}+2\theta +3\right)\right]}$
The first derivative of the log likelihood function is given by
$\frac{d\mathrm{log}L}{d\theta}=\frac{3n}{\theta}\frac{2n\theta}{{\theta}^{2}+2}\frac{n\left(\overline{x}+3\right)}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{2\left(\theta +1\right){f}_{x}}{\left[{x}^{2}+x+\left({\theta}^{2}+2\theta +3\right)\right]}}$
where
$\overline{x}$
is the sample mean.
The maximum likelihood estimate (MLE),
$\widehat{\theta}$
of
$\theta $
of PAD (1.1) is the solution of the equation
$\frac{d\mathrm{log}L}{d\theta}=0$
and is thus given by the solution of the nonlinear equation
$\frac{3n}{\theta}\frac{2n\theta}{{\theta}^{2}+2}\frac{n\left(\overline{x}+3\right)}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{2\left(\theta +1\right){f}_{x}}{\left[{x}^{2}+x+\left({\theta}^{2}+2\theta +3\right)\right]}}=0$
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 to solve above nonlinear equation to get maximum likelihood estimate of the parameter.
Method of moment estimate (MOME) of the Parameter:
Let
$\left({x}_{1},{x}_{2},\mathrm{...},{x}_{n}\right)$
be a random sample of size
$n$
from the PAD (1.1). Equating the population mean to the corresponding sample mean, the MOME
$\tilde{\theta}$
of
$\theta $
of PAD (1.1) is the solution of the following cubic equation
$\overline{x}{\theta}^{3}{\theta}^{2}+2\overline{x}\theta 6=0$
where
$\overline{x}$
is the sample mean.
Applications and Goodness of Fit of Pad
When events seem to occur at random, Poisson distribution is a suitable statistical model. Examples of events where Poisson distribution is a suitable model includes the number of customers arriving at a service point, the number of telephone calls arriving at an exchange , the number of fatal traffic accidents per week in a given state, the number of radioactive particle emissions per unit of time, the number of meteorites that collide with a test satellite during a single orbit, the number of organisms per unit volume of some fluid, the number of defects per unit of some materials, the number of flaws per unit length of some wire, are some amongst others. Further, the conditions for using Poisson distribution are the independence of events and equality of mean and variance, which are rarely satisfied completely in biomedical science and thunderstorms due to the fact that the occurrences of successive events in biomedical science and thunderstorms are dependent. Negative binomial distribution is the appropriate choice for the situation where successive events are dependent but negative binomial distribution requires higher degree of overdispersion Johnson et al [6]. In biomedical science and thunderstorms, these conditions are not fully satisfied. Generally, the count data in biomedical science and thunderstorms are either overdispersed or underdispersed. The main reason for selecting PLD and PAD to fit count data from biomedical science and thunderstorms are that these two distributions are always overdispersed and PAD has some flexibility over PLD.
Applications in ecology
Ecology is the branch of biology which deals with the relations and interactions between organisms and their environment, including their organisms. Since the organisms and their environment in the nature are complex, dynamic, interdependent, mutually reactive and interrelated, ecology deals with the various principles which govern such relationship between organisms and their environment. Firstly Fisher et al. [7] discussed the applications of Logarithmic series distribution (LSD) to model count data in the science of ecology. Later, Kempton [8] who fitted the generalized form of Fisher’s Logarithmic series distribution (LSD) to model insect data and concluded that it gives a superior fit as compared to ordinary Logarithmic series distribution (LSD). He also concluded that it gives better explanation for the data having exceptionally long tail. Tripathi & Gupta [9] proposed another generalization of the Logarithmic series distribution (LSD) which is flexible to describe shorttailed as well as longtailed data and fitted it to insect data and found that it gives better fit as compared to ordinary Logarithmic series distribution. Shanker [10], Mishra & Shanker [11] have discussed applications of generalized logarithmic series distributions (GLSD) to models data in ecology. Shanker & Hagos [12] have tried to fit PLD for data relating to ecology and observed that PLD gives satisfactory fit.
In this section we have tried to fit Poisson distribution (PD), Poisson Lindley distribution (PLD) and PoissonAkash distribution (PAD) to many count data from biological sciences using maximum likelihood estimates. The data were on haemocytometer yeast cell counts per square, on European red mites on apple leaves and European corn borers per plant.
It is obvious from above tables that in Table 4.1.1, PD gives better fit than PLD and PSD; in Table 4.1.2 PAD gives better fit than PD and PLD while in Table 4.1.3, PLD gives better fit than PD and PAD.
Applications in genetics
Genetics is the branch of biological science which deals with heredity and variation. Heredity includes those traits or characteristics which are transmitted from generation to generation, and is therefore fixed for a particular individual. Variation, on the other hand, is mainly of two types, namely hereditary and environmental. Hereditary variation refers to differences in inherited traits whereas environmental variations are those which are mainly due to environment. Much quantitative works seem to be done in genetics but so far no works has been done on fitting of PAD for count data in genetics. The segregation of chromosomes has been studied using statistical tool, mainly chisquare (
${\chi}^{2}$
). In the analysis of data observed on chemically induced chromosome aberrations in cultures of human leukocytes, Loeschke & Kohler [13] suggested the negative binomial distribution while Janardan & Schaeffer [14] suggested modified Poisson distribution. Shanker [10], Mishra & Shanker [11] have discussed applications of generalized Logarithmic series distributions (GLSD) to model data in mortality, ecology and genetics. Shanker & Hagos [12] have detailed study on the applications of PLD to model data from genetics. In this section an attempt has been made to fit to data relating to genetics using PAD, PLD and PD using maximum likelihood estimate. Also an attempt has been made to fit PAD, PLD, and PD to the data of Catcheside et al. [15,16] in Tables 4.2.2, 4.2.3, and 4.2.4.
Number of Yeast Cells Per Square 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
213 
202.1 
234 
236.8 
1 
128 
138.0 
99.4 
95.6 
2 
37 
47.1 
40.5 
39.9 
3 
18 
$\begin{array}{l}10.7\\ 1.8\\ 0.2\\ 0.1\end{array}\}$

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

$\begin{array}{l}16.6\\ 6.7\\ 2.7\\ 1.7\end{array}\}$

4 
3 
5 
1 
6 
0 
Total 

400.0 
400.0 
400.0 
Estimate of Parameter 

$\widehat{\theta}=0.6825$

$\widehat{\theta}=1.950236$

$\widehat{\theta}=2.260342$

${\chi}^{2}$


10.08 
11.04 
14.68 
d.f. 

2 
2 
2 
pvalue 

0.0065 
0.004 
0.0006 
Table 4.1.1: Observed and expected number of Haemocytometer yeast cell counts per square observed by Gosset [18].
Number of Red Mites Per Leaf 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
38 
25.3 
35.8 
36.3 
1 
17 
29.1 
20.7 
20.1 
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
$\begin{array}{l}3.1\\ 1.6\\ 0.8\\ 0.6\end{array}\}$

6.1
$\begin{array}{l}3.2\\ 1.6\\ 0.8\\ 0.7\end{array}\}$

4 
3 
5 
2 
6 
1 
7+ 
0 
Total 
80 
80.0 
80.0 
80.0 
Estimate of Parameter 

$\widehat{\theta}=1.15$

$\widehat{\theta}=1.255891$

$\widehat{\theta}=1.620588$

${\chi}^{2}$


18.27 
2.47 
2.07 
d.f. 

2 
3 
3 
pvalue 

0.0001 
0.4807 
0.558 
Table 4.1.2: Observed and expected number of red mites on Apple leaves.
Number of Corn Borer Per Plant 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
188 
169.4 
194.0 
196.3 
1 
83 
109.8 
79.5 
76.5 
2 
36 
35.6 
31.3 
30.8 
3 
14 
$\begin{array}{l}7.8\\ 1.2\\ 0.2\end{array}\}$

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

$\begin{array}{l}12.4\\ 4.9\\ 3.1\end{array}\}$

4 
2 
5 
1 
Total 
324 
324.0 
324.0 
324.0 
Estimate of Parameter 

$\widehat{\theta}=0.648148$

$\widehat{\theta}=2.043252$

$\widehat{\theta}=2.345109$

${\chi}^{2}$


15.19 
1.29 
2.33 
d.f. 

2 
2 
2 
pvalue 

0.0005 
0.5247 
0.3119 
Table 4.1.3: Observed and expected number of European cornborer of Mc Guire et al. [19].
It is obvious from the fitting of PAD, PLD, and PD that PAD gives much closer fit in Tables 4.2.1, 4.2.2 and 4.2.3 but in Table 4.2.4, PLD better fit than PD and PAD.
Number of Aberrations 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
268 
231.3 
257.0 
260.4 
1 
87 
126.7 
93.4 
89.7 
2 
26 
34.7 
32.8 
32.1 
3 
9 
$\begin{array}{l}6.3\\ 0.8\\ 0.1\\ 0.1\\ 0.1\end{array}\}$

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

11.5
$\begin{array}{l}4.1\\ 1.4\\ 0.5\\ 0.3\end{array}\}$

4 
4 
5 
2 
6 
1 
7+ 
3 
Total 
400 
400.0 
400.0 
400.0 
Estimate of Parameter 

$\widehat{\theta}=0.5475$

$\widehat{\theta}=2.380442$

$\widehat{\theta}=2.659408$

${\chi}^{2}$


38.21 
6.21 
4.17 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.1018 
0.2437 
Table 4.2.1: Distribution of number of Chromatid aberrations (0.2 g chinon 1, 24 hours).
Class/Exposure
$(\mu gkg)$

Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
413 
374.0 
405.7 
409.5 
1 
124 
177.4 
133.6 
128.7 
2 
42 
42.1 
42.6 
42.1 
3 
15 
$\begin{array}{l}6.6\\ 0.8\\ 0.1\\ 0.0\end{array}\}$

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

13.9
$\begin{array}{l}4.6\\ 1.5\\ 0.7\end{array}\}$

4 
5 
5 
0 
6 
2 
Total 
601 
601.0 
601.0 
601.0 
Estimate of Parameter 

$\widehat{\theta}=0.47421$

$\widehat{\theta}=2.685373$

$\widehat{\theta}=2.915059$

${\chi}^{2}$


48.17 
1.34 
0.29 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.7196 
0.9619 
Table 4.2.2: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure60
$\mu gkg$
.
Class/Exposure
$(\mu gkg)$

Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
200 
172.5 
191.8 
194.1 
1 
57 
95.4 
70.3 
67.6 
2 
30 
26.4 
24.9 
24.5 
3 
7 
$\begin{array}{l}4.9\\ 0.7\\ 0.1\\ 0.0\end{array}\}$

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

$\begin{array}{l}8.9\\ 3.2\\ 1.1\\ 0.6\end{array}\}$

4 
4 
5 
0 
6 
2 
Total 
300 
300.0 
300.0 
300.0 
Estimate of Parameter 

$\widehat{\theta}=0.55333$

$\widehat{\theta}=2.353339$

$\widehat{\theta}=2.626739$

${\chi}^{2}$


29.68 
3.91 
3.12 
d.f. 

2 
2 
2 
pvalue 

0.0000 
0.1415 
0.2101 
Table 4.2.3: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure70
$\mu gkg$
.
Class/Exposure
$(\mu gkg)$

Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
155 
127.8 
158.3 
160.7 
1 
83 
109.0 
77.2 
74.3 
2 
33 
46.5 
35.9 
35.3 
3 
14 
$\begin{array}{l}13.2\\ 2.8\\ 0.5\\ 0.2\end{array}\}$

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

16.5
$\begin{array}{l}7.5\\ 3.3\\ 2.4\end{array}\}$

4 
11 
5 
3 
6 
1 
Total 
300 
300.0 
300.0 
300.0 
Estimate of Parameter 

$\widehat{\theta}=0.853333$

$\widehat{\theta}=1.617611$

$\widehat{\theta}=1.963313$

${\chi}^{2}$


24.97 
1.51 
1.98 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.6799 
0.5766 
Table 4.2.4: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure90
$\mu gkg$
.
Applications in thunderstorms
In thunderstorm activity, the occurrence of successive thunderstorm events (THE’s) is often dependent process which means that the occurrence of a THE indicates that the atmosphere is unstable and the conditions are favorable for the formation of further thunderstorm activity. The negative binomial distribution (NBD) is a possible alternative to the Poisson distribution when successive events are possibly dependent Johnson et al. [6]. The theoretical and empirical justification for using the NBD to describe THE activity has been fully explained and discussed by Falls et al. [17]. Further, for fitting Poisson distribution to the count data equality of mean and variance should be satisfied. Similarly, for fitting NBD to the count data, mean should be less than the variance. In THE, these conditions are not fully satisfied. As a model to describe the frequencies of thunderstorms (TH’s), given an occurrence of THE, the PAD can be considered because it is always overdispersed Tables 4.3.1, 4.3.2, 4.3.3 and 4.3.4.
No. of Thunderstorms 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
187 
155.6 
185.3 
187.9 
1 
77 
117.0 
83.5 
80.2 
2 
40 
43.9 
35.9 
35.3 
3 
17 
$\begin{array}{l}11.0\\ 2.1\\ 0.3\\ 0.1\end{array}\}$

15.0 
15.4 
4 
6 
$\begin{array}{l}6.1\\ 2.5\\ 1.7\end{array}\}$

$\begin{array}{l}6.6\\ 2.7\\ 1.9\end{array}\}$

5 
2 
6 
1 
Total 
330 
330.0 
330.0 
330.0 
ML estimate 

$\widehat{\theta}=0.751515$

$\widehat{\theta}=1.804268$

$\widehat{\theta}=2.139736$

${\chi}^{2}$


31.93 
1.43 
1.35 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.6985 
0.7173 
Table 4.3.1: 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. [16].
No. of Thunderstorms 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
177 
142.3 
177.7 
180.0 
1 
80 
124.4 
88.0 
84.7 
2 
47 
54.3 
41.5 
40.9 
3 
26 
$\begin{array}{l}15.8\\ 3.5\\ 0.7\end{array}\}$

18.9 
19.4 
4 
9 
$\begin{array}{l}8.4\\ 6.5\end{array}\}$

$\begin{array}{l}8.9\\ 7.1\end{array}\}$

5 
2 
Total 
341 
341.0 
341.0 
341.0 
ML estimate 

$\widehat{\theta}=0.873900$

$\widehat{\theta}=1.583536$

$\widehat{\theta}=1.938989$

${\chi}^{2}$


39.74 
5.15 
5.02 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.1611 
0.1703 
Table 4.3.2: 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. [16].
No. of Thunderstorms 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
185 
151.8 
184.8 
187.5 
1 
89 
122.9 
87.2 
83.9 
2 
30 
49.7 
39.3 
38.6 
3 
24 
$\begin{array}{l}13.4\\ 2.7\\ 0.5\end{array}\}$

17.1 
17.5 
4 
10 
$\begin{array}{l}7.3\\ 5.3\end{array}\}$

$\begin{array}{l}7.6\\ 5.9\end{array}\}$

5 
3 
Total 
341 
341.0 
341.0 
341.0 
ML estimate 

$\widehat{\theta}=0.809384$

$\widehat{\theta}=1.693425$

$\widehat{\theta}=2.038417$

${\chi}^{2}$


49.49 
5.03 
4.69 
d.f. 

2 
3 
3 
pvalue 

0.0000 
0.1696 
0.196 
Table 4.3.3: 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. [16].
No. of Thunderstorms 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PAD 
0 
549 
449.0 
547.5 
555.1 
1 
246 
364.8 
259.0 
249.2 
2 
117 
148.2 
116.9 
114.9 
3 
67 
40.1 
51.2 
52.3 
4 
25 
$\begin{array}{l}8.1\\ 1.3\\ 0.5\end{array}\}$

21.9 
23.2 
5 
7 
$\begin{array}{l}9.2\\ 6.3\end{array}\}$

$\begin{array}{l}10.0\\ 7.3\end{array}\}$

6 
1 
Total 
1012 
1012.0 
1012.0 
1012.0 
ML estimate 

$\widehat{\theta}=0.812253$

$\widehat{\theta}=1.688990$

$\widehat{\theta}=2.033715$

${\chi}^{2}$


119.45 
9.60 
9.40 
d.f. 

3 
4 
4 
pvalue 

0.0000 
0.0477 
0.0518 
Table 4.3.4: 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. [16].
It is obvious from the fitting of PAD, PLD, and PD that PAD gives much closer fit than PLD and PD in all datasets relating to thunderstorms and hence PAD can be considered as an important model for modeling thunderstorms events.
References
 Shanker R (2016) The discrete PoissonAkash distribution. Communicated.
 Shanker R (2015) Akash distribution and Its Applications. International Journal of Probability and Statistics 4(3): 6575.
 Shanker R, Hagos F, Sujatha S (2016) On Modeling of Lifetime Data
Using One Parameter Akash, Lindley and Exponential Distributions.
Biometrics & Biostatistics International Journal 3(2): 110.
 Sankaran M (1970) The discrete PoissonLindley distribution. Biometrics 26(1): 145149.
 Lindley DV (1958) Fiducial distributions and Bayes theorem. Journal of Royal Statistical Society 20(1): 102107.
 Johnson NL, Kotz S, Kemp AW (1992) Univariate Discrete Distributions, 2nd edition. John Wiley & sons Inc, USA.
 Fisher RA, Corpet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology 12(1): 4258.
 Kempton RA (1975) A generalized form of Fisher’s logarithmic series. Biometrika 62(1): 2938.
 Tripathi RC, Gupta RC (1985) A generalization of the logseries distribution. Communications in Statistics 14 (8): 17791799.
 Shanker R (2002) Generalized Logarithmic Series Distributions and Their Applications, Unpublished Ph.D Thesis, Patna, India.
 Mishra A, Shanker R (2002) Generalized logarithmic series distributionIts nature and applications, Proceedings of the Vth International Symposium on Optimization and Statistics. 155168.
 Shanker R, Hagos F (2015) On PoissonLindley distribution and Its applications to Biological Sciences. Biometrics and Biostatistics International Journal 2(4): 15.
 Loeschke V, Kohler W (1976) Deterministic and Stochastic models of the negative binomial distribution and the analysis of chromosomal aberrations in human leukocytes. Biometrische Zeitschrift 18(6): 427451.
 Janardan KG, Schaeffer DJ (1977) Models for the analysis of chromosomal aberrations in human leukocytes. Biometrical Journal 19(8): 599612.
 Catcheside DG, Lea DE, Thoday JM (1946 a) Types of chromosome structural change induced by the irradiation on Tradescantia microspores. J Genet 47: 113136.
 Catcheside DG, Lea DE, Thoday JM (1946 b) The production of chromosome structural changes in Tradescantia microspores in relation to dosage, intensity and temperature. J Genet 47: 137149.
 Falls LW, Williford WO, Carter MC (1970) Probability distributions for thunderstorm activity at Cape Kennedy, Florida. Journal of Applied Meteorology 10: 97104.
 Gosset WS (1908) The probable error of a mean. Biometrika 6: 125.
 Mc Guire JU, Brindley TA, Bancroft TA (1957) The distribution of European cornborer larvae pyrausta in field corn. Biometrics 13(1): 6578.