Abstract
It is a compound discrete probability distribution which has two parameters and whose particular case is PoissonNew LinearExponential Distribution (PNLED) [1]. The statistical characteristics required for this distribution such as probability mass function (pmf), statistical moments, estimation of parameters and goodness of fit have been derived and explained nicely. To test theoretical reliability of this distribution, goodness of fit has been applied to some overdispersed data which were used by other researchers and further, we found that this distribution looks more appropriate for statistical modelling than PNLED, generalised PoissonLindley distribution (GPLD) [1, 2] and PoissonLindley distribution (PLD) [3].
1. Introduction
A generalisation of any probability distribution is considered more appropriate when it is more reliable than its particular case as well as previously obtained generalised distribution with similar conditions in all aspects.
The proposed distribution is a compound distribution. For compounding process, at least two distributions are required. We have needed Generalised Poisson distribution (GPD) [4] and New Linearexponential distribution (NLED) [5] to conduct compounding process. GPD [4] distribution is a discrete distribution whereas NLED [5] is a continuous distribution. GPD [4] has two parameters and in mixing process it plays role of an original distribution and its original parameter $\lambda $ follows NLED [5]. The pmf of GPD [4] has been given by:
where $Z=\mathrm{0,1},2,\dots $; $\lambda >0$; $\left\theta \right<1$.
The probability density function (pdf) of NLED [5] was given by:
Hence, Generalised PoissonNew Linearexponential distribution (GPNLED) has been obtained by compounding GPD [4] with NLED [5]. The study of this papers mentioned in the following references [1, 2], [4], [6] have very much contributed to increase quality of the paper.
In this paper, the following works have been done. Construction and derivation of:
1) Probability Mass Function (pmf) of GPNLED.
2) Statistical Moments of GPNLED.
3) Estimation of Parameters of GPNLED.
4) Goodness of Fit and applications of GPNLED.
We have been studied the following papers to improve quality of this paper which have been mentioned in the following references [2], [714].
2. Results
The results obtained for this paper have been placed under the following subheadings.
2.1. Probability mass function of GPNLED
GPD has two parameters $\lambda $ and $\theta $. $\lambda $ is an original parameter of GPD which follows NLED. $\theta $ is an additional parameter of GPD which is versatile in nature and hence, GPD. Here, $\lambda >0$ and $\theta $ < 1. NLED has a single parameter $\beta $. The Probability mass of function of GPNLED is obtained by mixing GPD with NLED. This is how the pmf of GPNLED have been extracted:
$=\left[\frac{{\beta}^{2}{e}^{\theta z}}{z!(1+\pi \beta )}\right]\left\{\sum _{i=0}^{z1}\frac{(z1)!}{i!(zi1)!}{\left(\frac{\theta z}{1}\right)}^{i}\left(\frac{\pi \mathrm{\Gamma}(zi+1)}{(1+\beta {)}^{zi+1}}+\frac{\pi \mathrm{\Gamma}(zi+2)}{(1+\beta {)}^{zi+2}}\right)\right\}$
$=\left[\frac{{\beta}^{2}{e}^{\theta z}}{z(1+\pi \beta )}\right]\left\{\sum _{i=0}^{z1}\left(\frac{{\theta}^{i}{z}^{i}}{i!(zi1)!}\right)\left(\frac{\mathrm{\Gamma}(zi+1)}{(1+\beta {)}^{zi+1}}\right)\left(\pi +\frac{(zi+1)}{(1+\beta )}\right)\right\}$
$=\left[\frac{{\beta}^{2}{e}^{\theta z}}{(1+\pi \beta )}\right]\left\{\sum _{i=0}^{z1}\left(\frac{{\theta}^{i}{z}^{i1}}{i!}\right)\left(\frac{(zi)}{(1+\beta {)}^{zi+2}}\right)\left(\pi (1+\beta )+(zi+1)\right)\right\}$
$=\left[\frac{{\beta}^{2}{e}^{\theta z}}{(1+\pi \beta )}\right]\left[\frac{\left\{\pi (1+\beta )+(z+1)\right\}}{(1+\beta {)}^{z+2}}\right]+\left[\frac{{\beta}^{2}{e}^{\theta z}}{(1+\pi \beta )}\right]\left\{\sum _{i=1}^{z1}\left(\frac{{\theta}^{i}z(zi{)}^{i1}}{i!}\right)\left(\frac{\left\{\pi (1+\beta )+(zi+1)\right\}}{(1+\beta {)}^{zi+2}}\right)\right\}$
$z=\mathrm{0,1},2,...,\lambda >0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\beta >0.$
Probability mass function of GPNLED can be obtained for each value such as $z=\mathrm{0,1},2,...$ can be obtained by using the Eq. (3) as follows:
$\left.+\frac{16{\theta}^{3}\left\{\pi (1+\beta )+2\right\}}{6(1+\beta {)}^{3}}\right],$
$\left.+\frac{50{\theta}^{3}\left\{\pi (1+\beta )+3\right\}}{6(1+\beta {)}^{4}}+\frac{125{\theta}^{4}\left\{\pi (1+\beta )+2\right\}}{24(1+\beta {)}^{3}}\right].$
2.2. Statistical moments of GPNLED
Let ${{\mu}_{r}}^{\text{'}}$ denotes $r$th moment about origin of GPNLED:
Fig. 1Showing the mean of GPNLED
Central moments of GPNLED:
Fig. 2Showing the variance of GPNLED
Proof. Variance > Mean.
Or:
Or:
Or$:$
Or:
Hence the statement.
The third moment about the mean of this distribution can be obtained as:
$=\frac{\left[\begin{array}{c}[\left\{{\beta}^{2}(1+\pi \beta {)}^{2}(1+2\theta \left)\right(\pi \beta +2)\right\}+\left\{3(1\theta )\beta (1+\pi \beta {)}^{2}(2\pi \beta +6)\right\}\\ +\left\{(1\theta {)}^{2}(1+\pi \beta {)}^{2}(6\pi \beta +24)\right\}]\\ [\left\{3(1\theta )\beta (1+\pi \beta )(\pi \beta +2{)}^{2}\right\}\left\{6(1\theta {)}^{2}(1+\pi \beta \left)\right(\pi \beta +2\left)\right(\pi \beta +3)\right\}\\ +\left[\left\{2(\pi \beta +2{)}^{3}\right\}\right]\end{array}\right]}{\left[\beta \right(1+\pi \beta \left){]}^{3}\right(1\theta {)}^{5}}.$
The third central moment (${\mu}_{3}$) gives positive value, this distribution is positively skewed in shape.
$\left.+\frac{6\left(\pi \beta +4\right)}{{\beta}^{3}(1+\pi \beta )(1\theta {)}^{5}}\right]\left[\frac{\left(\pi \beta +2\right)}{\beta \left(1+\pi \beta \right)\left(1\theta \right)}\right]+6\left[\frac{\left[\beta \left(\pi \beta +2\right)+2\left(1\theta \right)\left(\pi \beta +3\right)\right]}{{\beta}^{2}(\pi \beta +1)(1\theta {)}^{3}}\right]{\left[\frac{\left(\pi \beta +2\right)}{\beta \left(1+\pi \beta \right)\left(1\theta \right)}\right]}^{2}3{\left[\frac{\left(\pi \beta +2\right)}{\beta \left(1+\pi \beta \right)\left(1\theta \right)}\right]}^{4}$
The Eq. (18) is obtained to know about nature of distribution according to size and it is the fourth central moment. Coefficient of skewness and kurtosis based on moments can be obtained as follows.
From Eq. (19), it has been found that range of ${\gamma}_{1}$ is $\left(\sqrt{2}\right)<{\gamma}_{1}<\infty )$. Hence, it is positively skewed:
2.3. Estimation of parameters
This distribution has two parameters $\beta $ and $\theta $ which can be obtained as
By using $P(z=0)$ and ${{\mu}_{1}}^{\text{'}}$. We have:
Solving it, we get:
The Polynomial Eq. (21) can be solved by using RegulaFalshi method or Newton Rapson method, where $k$ denote $P(z=0)$.Substituting the estimated value of $\beta $ in the following Eq. (22), we get an estimated value of $\theta $:
${{\mu}_{1}}^{\text{'}}$ and ${{\mu}_{2}}^{\text{'}}$: substituting the value of $(1\theta )$ obtained from Eq. (22) in the expression of ${{\mu}_{2}}^{\text{'}}$, we get Polynomial Eq. (23) in $\beta $ which can be solve by NewtonRapson or ReglaFalsi method:
2.4. Goodness of fit and applications of GPNLED
This distribution can be used to test goodness of fit and to get possible inferential Statistics in the field related to accident proneness, risk management of production engineering, ecological sciences, agricultural fields related to events about insects, error per page, biological sciences, and other fields [1517].
Applications of GPNLE model in the field of engineering. This distribution is applicable to fit better where Poisson, Generalised Poisson and Poisson mixtures of continuous distributions are applicable such as:
1) The count of $\alpha $particles emitted per unit of time is useful in analysis of any radioactive substances.
2) Number of telephone calls received on a given switch board per small unit of time.
3) In industrial production to find the proportion of defects per unit length, per unit area, etc.
4) In the field of reliability engineering.
In the following examples Chisquare goodness of fit have been applied by using GPNLE model.
Table 1Example (1)
Z  0  1  2  3  ${4}^{+}$ 
Observed frequency  35  11  8  4  2 
Table 2Example (2)
Number of accidents  0  1  2  3  4  5^{+} 
Observed frequency  447  132  42  21  3  2 
Table 3Example (3)
Z  0  1  2  3  4  5  6 
Observed frequency  200  57  30  7  4  0  2 
The first example is related to the number of errors per page which was given by Kemp and Kemp [18]. The second example is related to the data of accidents to 647 women working on H. E. Shells in 5 weeks was given by Greenwood and Yule [19]. The data related to Class per exposure (μg/kg) which is included in example (3) was given by Catcheside et al. [20].
The first and second examples have been used in the Doctoral Thesis. The theoretical frequencies obtained by using PLD, GPLD and GPNLED have been placed in Table 4 and 5 to make comparison. In Table 6, the theoretical frequencies obtained by using PLD, PNLED and GPNLED have been placed for comparison [13].
Table 4Observed versus expected frequency of example (1)
Number of errors per page  Observed frequency  Expected frequency  
PLD  GPLD  GPNLED  
0  35  33.0  35.0  35.0 
1  11  15.3  13.3  13.3 
2  8  6.8  6.3  6.4 
3  4  2.9  3.0  2.9 
4^{+}  2  2.0  2.4  2.4 
60.0  60.0  60.0  647.0  
${{\mu}_{1}}^{\text{'}}$  0.7833333  –  –  – 
${{\mu}_{2}}^{\text{'}}$  1.85  –  –  – 
$\widehat{\beta}$  –  1.7434  1.9387733  1.642188744 
$\widehat{\theta}$  –  –  0.1174867  0.09640955 
Table 5Observed versus expected frequency of example (2)
Number of accidents  Observed frequency  Expected frequency  
PLD  GPLD  GPNLED  
0  447  441  447.0  447.0 
1  132  143  1132.7  132.9 
2  42  45  44.8  45.0 
3  21  14  15.1  15.0 
4  3  4  5.0  5.0 
5^{+}  2  1  2.4  2.1 
647  647  647.0  647.0  
${{\mu}_{1}}^{\text{'}}$  0.4652241  –  –  – 
${{\mu}_{2}}^{\text{'}}$  0.9100646  –  –  – 
$\widehat{\beta}$  –  –  2.8563455  2.496217 
$\widehat{\theta}$  –  –  0.0523222  0.0415098 
Table 6Observed versus expected frequency of example (3)
Class per exposure (μg/kg)  Observed frequency  Expected frequency  
PLD  PNLED  GPNLED  
0  200  191.8  192.8  200.0 
1  57  70.3  69.2  60.5 
2  30  24.9  24.6  23.7 
3  7  8.6  8.7  9.5 
4  4  2.9  3.1  3.8 
5  0  1.0  1.1  1.5 
6  2  0.5  0.5  1.0 
300  300.0  300.0  300.0  
${{\mu}_{1}}^{\text{'}}$  0.55333333  –  –  – 
${{\mu}_{2}}^{\text{'}}$  1.2533333  –  –  – 
$\widehat{\beta}$  –  2.353339  2.0501155  2.256957115 
$\widehat{\theta}$  –  –  –  0.1002898421 
3. Conclusions
1) This distribution will be overdispersed if $(1\theta )\le \frac{({\pi}^{2}{\beta}^{2}+4\pi \beta +2)}{({\pi}^{2}{\beta}^{3}+3\pi {\beta}^{2}+2\beta )}$.
2) The range of ${\gamma}_{1}$ is $\left(\sqrt{2}\right)<{\gamma}_{1}<\infty )$. So, it is positively skewed in shape.
3) The range of ${\beta}_{2}$ is $6<{\beta}_{2}<\infty $. So, it is leptokurtic by size.
4) From the Tables 4 and 5, it is found that Pvalue obtained by using GPNLED is greater than those obtained by using GPLD [2] and PLD [3]
5) From Table 6, what we have observed that the Pvalue obtained by using GPNLED is bigger than those of PNLED [1] and PLD [3]
6) Hence, it is suggested to apply GPNLED instead of PLD, GPLD and PNLED in similar situation in all aspects.
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About this article
The authors have not disclosed any funding.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Binod Kumar Sah: conceptualization, data curation, formal analysis, investigation, writing – original draft preparation. Suresh Kumar Sahani: conceptualization, data curation, formal analysis, investigation, supervision, writing – original draft preparation, writing – review and editing.
The authors declare that they have no conflict of interest.