In this paper some important properties including coefficients of variation, skewness, kurtosis and index of dispersion of size–biased new quasi Poisson–Lindley distribution (SBNQPLD) have been discussed and their behaviors have been explained graphically for varying values of parameters. Some applications of SBNQPLD have also been discussed.

Keywords: Size–biased new Quasi Poisson–Lindley distribution, moments based measures, maximum likelihood estimation, goodness of fit

The size– biased version of Poisson–Lindley distribution (SBPLD) proposed by Sankaran¹ has been introduced by Ghitany and Mutairi² and is defined by its probability mass function (pmf)

$P_1\left(x,\theta \right)=\frac{{\theta }^3}{\theta +2}\cdot \frac{x\left(x+\theta +2\right)}{{\left(\theta +1\right)}^{x+2}};\theta >0,x=1,2,3,\mathrm{...}$

Shanker et al.³ have proposed a simple method of deriving moments of SBPLD and the applications of SBPLD to model thunderstorms events. The Poisson –Lindley distribution (PLD), a Poisson mixture of Lindley distribution of Lindley,⁴ is defined by its pmf

$P_2\left(x;\theta \right)=\frac{{\theta }^2\left(x+\theta +2\right)}{{\left(\theta +1\right)}^{x+3}};x=0,1,2,\dots , >0.$                                     

The Lindley distribution is defined by its probability density function (pdf)                        $f_1\left(x,\theta \right)=\frac{{\theta }^2}{\theta +1}\left(1+x\right)e^{-\theta x}$ ; x > 0, $\theta$ > 0                                                           

The size–biased quasi Poisson–Lindley distribution (SBQPLD), size–biased version of quasi Poisson–Lindley distribution (QPLD) of Shanker and Mishra,⁵ suggested by Shanker and Mishra⁶ with parameters $\theta$ and $\alpha$ is defined by its pmf

$P_3\left(x;\theta ,\alpha \right)=\frac{{\theta }^2}{\alpha +2}\frac{x\left(\theta x+\theta \alpha +\theta +\alpha \right)}{{\left(\theta +1\right)}^{x+2}};x=1,2,3,\mathrm{...},\theta >0,\alpha >-2$         

The QPLD, a Poisson mixture of quasi Lindley distribution proposed by Shanker and Mishra,⁷ is defined by its pmf

$P_4\left(x;\theta ,\alpha \right)=\frac{\theta \left(\theta x+\theta \alpha +\theta +\alpha \right)}{\left(\alpha +1\right){\left(\theta +1\right)}^{x+2}};x=0,1,2,\mathrm{...};\theta >0,\alpha >-1$          

The QLD is defined by its pdf

$f_2\left(x;\theta ,\alpha \right)=\frac{\theta }{\alpha +1}\left(\alpha +x\theta \right)e^{-\theta x};x>0,\theta \text{ > 0, }\alpha >-1$                        

Shanker and Amanuel⁸ proposed a new quasi Lindley distribution (NQLD) having pdf

$f_3\left(x;\theta ,\alpha \right)=\frac{{\theta }^2}{{\theta }^2+\alpha }\left(\theta +\alpha x\right)e^{-\theta x}$

where$\theta +\alpha x>0\text{and}{\theta }^2+\alpha >0\text{or}\theta +\alpha x<0\text{and}{\theta }^2+\alpha <0$ for $x>0$ $\theta >0$ . Lindley distribution is a particular case of NQLD at $\alpha =\theta$ . A new quasi Poisson–Lindley distribution (NQPLD), a Poisson mixture of NQLD, has been suggested by Shanker and Tekie⁹ and defined by its pmf

$P_5\left(x;\theta ,\alpha \right)=\frac{{\theta }^2}{{\left(\theta +1\right)}^{x+2}}\left[1+\frac{\theta +\alpha x}{{\theta }^2+\alpha }\right]$

where $\theta +\alpha x>0\text{and}{\theta }^2+\alpha >0\text{or}\theta +\alpha x<0\text{and}{\theta }^2+\alpha <0$ for $x=0,1,2,\mathrm{...};$ $\theta >0$ .

It can be seen that the PLD is a particular case of it at $\alpha =\theta$ . Shanker et al.¹⁰ derived the pmf of size biased new quasi Poisson–Lindley distribution (SBNQPLD) having pmf

$P_6\left(x;\theta ,\alpha \right)=\frac{{\theta }^3}{{\theta }^2+2\alpha }\frac{x\left({\theta }^2+\theta +\alpha +\alpha x\right)}{{\left(\theta +1\right)}^{x+2}};x=1,2,3,\mathrm{...},\theta >0,{\theta }^2+2\alpha >0$

Shanker et al.¹⁰ discussed various statistical properties, parameters estimation and applications of SBNQPLD. Shanker et al.¹⁰ have shown that SBNQPLD can also be obtained from the size–biased Poisson distribution when its parameter $\lambda$ follows a SBNQLD with pdf       

$f_4\left(\lambda ;\theta ,\alpha \right)=\frac{{\theta }^3}{{\theta }^2+2\alpha }\lambda \left(\theta +\alpha x\right)e^{-\theta x}$

 where $\theta +\alpha x>0\text{and}{\theta }^2+\alpha >0\text{or}\theta +\alpha x<0\text{and}{\theta }^2+\alpha <0$ for $x>0$ $\theta >0$ . That is

$P\left(X=x\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{e^{-\lambda }{\lambda }^{x-1}}{\left(x-1\right)!}}\cdot \frac{{\theta }^3}{{\theta }^2+2\alpha }\lambda \left(\theta +\alpha \lambda \right)e^{-\theta \lambda }d\lambda$

$=\frac{{\theta }^3}{{\theta }^2+2\alpha }\frac{x\left({\theta }^2+\theta +\alpha +\alpha x\right)}{{\left(\theta +1\right)}^{x+2}}$ $;x=1,2,3,\mathrm{...}$

The $r$ th factorial moment about origin ${\mu }_{\left(r\right)}{}^{\prime }$ of SBNQPLD obtained by Shanker et al.¹⁰ as

${\mu }_{\left(r\right)}{}^{\prime }=\frac{r!\left\{r{\theta }^3+\left(r+1\right){\theta }^2+r\left(r+1\right)\alpha \theta +\left(r+1\right)\left(r+2\right)\alpha \right\}}{{\theta }^r\left({\theta }^2+2\alpha \right)};r=1,2,3,\mathrm{...}$

Thus, the first four moments about origin obtained by Shanker et al.¹⁰ are

${\mu }_1{}^{\prime }=1+\frac{2\left({\theta }^2+3\alpha \right)}{\theta \left({\theta }^2+2\alpha \right)}$

${\mu }_2{}^{\prime }=1+\frac{6\left({\theta }^2+3\alpha \right)}{\theta \left({\theta }^2+2\alpha \right)}+\frac{6\left({\theta }^2+4\alpha \right)}{{\theta }^2\left({\theta }^2+2\alpha \right)}$

${\mu }_3{}^{\prime }=1+\frac{14\left({\theta }^2+3\alpha \right)}{\theta \left({\theta }^2+2\alpha \right)}+\frac{36\left({\theta }^2+4\alpha \right)}{{\theta }^2\left({\theta }^2+2\alpha \right)}+\frac{24\left({\theta }^2+5\alpha \right)}{{\theta }^3\left({\theta }^2+2\alpha \right)}$

${\mu }_4{}^{\prime }=1+\frac{30\left({\theta }^2+3\alpha \right)}{\theta \left({\theta }^2+2\alpha \right)}+\frac{126\left({\theta }^2+4\alpha \right)}{{\theta }^2\left({\theta }^2+2\alpha \right)}+\frac{240\left({\theta }^2+5\alpha \right)}{{\theta }^3\left({\theta }^2+2\alpha \right)}+\frac{120\left({\theta }^2+6\alpha \right)}{{\theta }^4\left({\theta }^2+2\alpha \right)}$ .

It has been observed that the central moments (moments about the mean) has not been given by et al.¹⁰ and hence many important characteristics including coefficient of variation, skewness, kurtosis and index of dispersion of SBNQPLD has not been studied by Shanker et al.¹⁰

The main purpose of this paper is to derive expressions for coefficients of variation, skewness, kurtosis and index of dispersion of SBNQPLD and study their behaviour graphically. The goodness of fit of the distribution has been presented with a number of count datasets using maximum likelihood estimates from various fields of knowledge.    

Using the relationship between moments about the mean and the moments about the origin, the moments about mean of SBNQPLD can be obtained as

${\mu }_2=\frac{2\left({\theta }^5+{\theta }^4+5\alpha {\theta }^3+6\alpha {\theta }^2+6{\alpha }^2\theta +6{\alpha }^2\right)}{{\theta }^2{\left({\theta }^2+2\alpha \right)}^2}$

${\mu }_3=\frac{2\left\{\begin{array}{l}{\theta }^8+3{\theta }^7+\left(7\alpha +2\right){\theta }^6+24\alpha {\theta }^5+\left(16{\alpha }^2+18\alpha \right){\theta }^4+54{\alpha }^2{\theta }^3\\ +\left(12{\alpha }^3+36{\alpha }^2\right){\theta }^2+36{\alpha }^3\theta +24{\alpha }^3\end{array}\right\}}{{\theta }^3{\left({\theta }^2+2\alpha \right)}^3}$

${\mu }_4=\frac{2\left\{\begin{array}{l}{\theta }^{11}+13{\theta }^{10}+\left(9\alpha +24\right){\theta }^9+\left(130\alpha +12\right){\theta }^8+\left(30{\alpha }^2+264\alpha \right){\theta }^7\\ +\left(460{\alpha }^2+144\alpha \right){\theta }^6+\left(44{\alpha }^3+936{\alpha }^2\right){\theta }^5+\left(696{\alpha }^3+504{\alpha }^2\right){\theta }^4\\ +\left(24{\alpha }^4+1368{\alpha }^3\right){\theta }^3+\left(384{\alpha }^4+720{\alpha }^3\right){\theta }^2+720{\alpha }^4\theta +360{\alpha }^4\end{array}\right\}}{{\theta }^4{\left({\theta }^2+2\alpha \right)}^4}$ .

The coefficient of variation (C.V), coefficient of Skewness $\left(\sqrt{{\beta }_1}\right)$ , coefficient of Kurtosis $\left({\beta }_2\right)$ and Index of dispersion $\left(\gamma \right)$ of SBNQPLD are obtained as

$C.V.=\frac{\sigma }{{\mu }_1{}^{\prime }}=\frac{\sqrt{2\left({\theta }^5+{\theta }^4+5\alpha {\theta }^3+6\alpha {\theta }^2+6{\alpha }^2\theta +6{\alpha }^2\right)}}{{\theta }^3+2{\theta }^2+2\alpha \theta +6\alpha }$

$\sqrt{{\beta }_1}=\frac{{\mu }_3}{{\left({\mu }_2\right)}^{3/2}}=\frac{\left\{\begin{array}{l}{\theta }^8+3{\theta }^7+\left(7\alpha +2\right){\theta }^6+24\alpha {\theta }^5+\left(16{\alpha }^2+18\alpha \right){\theta }^4+54{\alpha }^2{\theta }^3\\ +\left(12{\alpha }^3+36{\alpha }^2\right){\theta }^2+36{\alpha }^3\theta +24{\alpha }^3\end{array}\right\}}{\sqrt{2}{\left({\theta }^5+{\theta }^4+5\alpha {\theta }^3+6\alpha {\theta }^2+6{\alpha }^2\theta +6{\alpha }^2\right)}^{3/2}}$

${\beta }_2=\frac{{\mu }_4}{{\mu }_2{}^2}=\frac{\left\{\begin{array}{l}{\theta }^{11}+13{\theta }^{10}+\left(9\alpha +24\right){\theta }^9+\left(130\alpha +12\right){\theta }^8+\left(30{\alpha }^2+264\alpha \right){\theta }^7\\ +\left(460{\alpha }^2+144\alpha \right){\theta }^6+\left(44{\alpha }^3+936{\alpha }^2\right){\theta }^5+\left(696{\alpha }^3+504{\alpha }^2\right){\theta }^4\\ +\left(24{\alpha }^4+1368{\alpha }^3\right){\theta }^3+\left(384{\alpha }^4+720{\alpha }^3\right){\theta }^2+720{\alpha }^4\theta +360{\alpha }^4\end{array}\right\}}{2{\left({\theta }^5+{\theta }^4+5\alpha {\theta }^3+6\alpha {\theta }^2+6{\alpha }^2\theta +6{\alpha }^2\right)}^2}$

$\gamma =\frac{{\sigma }^2}{{\mu }_1{}^{\prime }}=\frac{2\left({\theta }^5+{\theta }^4+5\alpha {\theta }^3+6\alpha {\theta }^2+6{\alpha }^2\theta +6{\alpha }^2\right)}{\theta \left({\theta }^2+2\alpha \right)\left({\theta }^3+2{\theta }^2+2\alpha \theta +6\alpha \right)}$ .

Shapes of coefficient of variation, skewness, kurtosis and index of dispersion of SBNQPLD for varying values of parameters have been shown in figure 1.

Figure 1 Behaviors of C.V, Skewness, Kurtosis and Index of dispersion of SBNQPLD for values of $\theta$ and $\alpha$

Suppose $\left(x_1,x_2,\dots ,x_n\right)$ as random samples of size n from the SBNQPLD and $f_x$ , the observed frequency in the sample corresponding to $X=x$ $\left(x=1,2,\mathrm{...},k\right)$ such that ${\displaystyle \sum _{x=1}^k{f}_x=n}$ , where $k$ being the largest observed value having non–zero frequency. The log likelihood function of SBNQPLD can be presented as

$\log L=n\log \left(\frac{{\theta }^3}{{\theta }^2+2\alpha }\right)-{\displaystyle \sum _{x=1}^k{f}_x\left(x+2\right)\log \left(\theta +1\right)}+{\displaystyle \sum _{x=1}^k{f}_x\log \left[\alpha x^2+x\left({\theta }^2+\theta +\alpha \right)\right]}$

The two log likelihood equations are thus obtained as

$\frac{\partial \log L}{\partial \theta }=\frac{3n}{\theta }-\frac{2n\theta }{{\theta }^2+2\alpha }-{\displaystyle \sum _{x=1}^k\frac{\left(x+2\right)f_x}{\theta +1}}+{\displaystyle \sum _{x=1}^k\frac{\left(2\theta +1\right)x{f}_x}{\left[\alpha x^2+x\left({\theta }^2+\theta +\alpha \right)\right]}}=0$

$\frac{\partial \log L}{\partial \alpha }=\frac{-2n}{{\theta }^2+2\alpha }+{\displaystyle \sum _{x=1}^k\frac{x\left(x+1\right)f_x}{\left[\alpha x^2+x\left({\theta }^2+\theta +\alpha \right)\right]}}=0$ .

These two log likelihood equations seems difficult to solve directly as these cannot be expressed in closed forms. The (MLE’s) $(\widehat{\theta },\widehat{\alpha })$ of parameters $(\theta ,\alpha )$ can be computed directly by solving the log likelihood equation using Newton–Raphson iteration available in R–software till sufficiently close values of $\widehat{\theta }\text{and}\widehat{\alpha }$ are obtained. The initial values of parameters θ and α are the MOME $(\tilde{\theta },\tilde{\alpha })$ of the parameters $(\theta ,\alpha )$ , given in Shanker et al.¹⁰

To test the goodness of fit of SBNQPLD along with SBPD, SBPLD and SBQPLD, several cont datasets have been considered from various fields of knowledge. The expected frequencies of SBPD, SBPLD and SBQPLD have also been given in the tables (Table 1–10). The estimates of the parameters have been obtained by the method of maximum likelihood. It is obvious from the goodness of fit of SBNQPLD that it provides better fit over SBPD and SBPLD and competing well with SBQPLD in majority of datasets. The following datasets have been considered for testing the goodness of fit of SBNQPLD.

In this paper expressions based ob central moments including coefficients of variation, skewness, kurtosis and index of dispersion of SBNQPLD have been derived and their behaviors have been explained graphically for varying values of the parameters. Some important applications of SBNQPLD have also been discussed and its goodness of fit has been compared with other discrete distributions. It has been observed that SBNQPLD provides much better fit over SBPD, SBPLD and competing well with SBQPLD in majority of datasets.

None.

None.

1. Sankaran M. The discrete Poisson–Lindley distribution, Biometrics, 1970;26(1):145–149.
2. Ghitany ME, Al–Mutairi DK. Size– biased Poisson–Lindley distribution and its applications. International Journal of Statistics. 2008; LXVI(3):299–311.
3. Shanker R, Hagos F, Abrehe Y. On size–biased Poisson–Lindley distribution and Its applications to model Thunderstorms, American Journal of Mathematics and Statistics. 2015;5(6):354–360.
4. Lindley DV. Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, Series B. 1958;20(1):102– 107.
5. Shanker R, Mishra A. A quasi Poisson–Lindley distribution. Journal of Indian Statistical Association. 2016;54(1&2):113–125.
6. Shanker R, Mishra A. On a size–biased quasi poisson lindley distribution. International Journal of Probability and Statistics, 2013a;2(2):28–34.
7. Shanker R, Mishra A. A quasi Lindley distribution. African journal of mathematics and computer science research. 2013b;6(4):64.
8. Shanker R, Amanuel AG. A new quasi Lindley distribution. International Journal of Statistics and Systems. 2013;8(2): 143–156.
9. Shanker R, Tekie AL. A new quasi Poisson–Lindley distribution, International Journal of Statistics and Systems. 2014;9(1):87–94.
10. Shanker R, Shanker R, Tekie AL. et al. Size–biased new quasi poisson lindley distribution and its applications. International Journal of Statistics and Systems. 2014;9(2):109–118.
11. Coleman JS, James J. The equilibrium size distribution of freely–forming groups. Sociometry. 1961;24(1):36–45.
12. Mathews JNS, Appleton DR. An application of the truncated Poisson distribution to Immunogold assay. Biometrics. 1993;49(2):617–621.
13. Keith LB, Meslow EC. Trap response by snowshoe hares. Journal of Wildlife Management. 1968;32(4):795–801.
14. Simonoff JS. Analyzing categorical data, New York Springer. 2003.
15. Finney DJ, Varley GC. An example of the truncated Poisson distribution, Biometrics. 1955;11:387–394.
16. Singh SN, Yadav RC. Trends in rural out–migration at household level, Rural Demogr. 1971;8(1):53–61.