In the present paper, firstly the nature of zero-truncated Poisson distribution (ZTPD), zero-truncated Poisson-Lindley distribution (ZTPLD) and zero-truncated Poisson-Sujatha distribution (ZTPSD) have been discussed with graphs of their probability functions for different values of their parameter. Over-dispersion, equi-dispersion and under-dispersion of ZTPLD and ZTPSD have been discussed using index of dispersion. A simple and interesting method of finding moments of ZTPSD has been suggested and thus the first two moments about origin and the variance have been obtained. The estimation of parameter of ZTPD, ZTPLD and ZTPSD has been discussed using maximum likelihood estimation and method of moments.

The goodness of fit of ZTPD, ZTPLD, and ZTPSD using maximum likelihood estimate in zero-truncated data arising from demography, biological sciences and migration has been discussed and observed that in most data-sets relating to demography and biological sciences, ZTPSD gives better fit than ZTPD and ZTPLD.

Keywords: zero-truncated distribution, poisson-lindley distribution, poissonsujatha distribution, moments, estimation of parameter, goodness of fit

Zero-truncated distributions, in probability theory, are certain discrete distributions having support the set of positive integers. These distributions are applicable for the situations when the data to be modeled originate from a mechanism that generates data excluding zero-counts.

Let $P_0\left(x;\theta \right)$  is the original distribution with support non negative positive integers. Then the zero-truncated version of $P_0\left(x;\theta \right)$  with the support the set of positive integers is given by

$P\left(x;\theta \right)=\frac{P_0\left(x;\theta \right)}{1-P_0\left(0;\theta \right)};x=1,2,3,\mathrm{...}$                                                        (1.1)

The Poisson-Lindley distribution (PLD) having probability mass function (p.m.f.)

$P_0\left(x;\theta \right)=\frac{{\theta }^2\left(x+\theta +2\right)}{{\left(\theta +1\right)}^{x+3}};x=0,1,2,3,\mathrm{...},\theta >0$                                  (1.2)

has been  introduced by Sankaran¹ to model count data. Recently, Shanker & Hagos² have done an extensive study on its applications to Biological Sciences and found that PLD provides a better fit than Poisson distribution to almost all biological science data. The PLD arises from the Poisson distribution when its parameter $\lambda$ follows Lindley³ with probability density function (p.d.f.)

$g\left(\lambda ;\theta \right)=\frac{{\theta }^2}{\theta +1}\left(1+\lambda \right)e^{-\theta \lambda };\lambda >0,\theta >0$                                (1.3)

Detailed study of Lindley distribution (1.3) has been done by Ghitany et al.⁴ and shown that (1.3) is a better model than exponential distribution for modeling some lifetime data. Recently, Shanker et al.⁵ showed that (1.3) is not always a better model than the exponential distribution for modeling lifetime’s data. In fact, Shanker et al.⁶ has done a very extensive and comparative study on modeling of lifetime data using exponential and Lindley distributions and discussed various lifetime data-sets to show the superiority of Lindley over exponential and that of exponential over Lindley distribution. The PLD has been extensively studied by Sankaran⁸ and Ghitany & Mutairi⁷ and its various properties have been discussed by them.  The Lindley distribution and the PLD have been generalized by many researchers.  Shanker & Mishra⁸ obtained a two parameter Poisson-Lindley distribution by compounding Poisson distribution with a two parameter Lindley distribution introduced by Shanker & Mishra.⁹ A quasi Poisson-Lindley distribution has been introduced by Shanker & Mishra¹⁰ by compounding Poisson distribution with a quasi Lindley distribution introduced by Shanker & Mishra.¹¹ Shanker et al.¹² obtained a discrete two parameter Poisson-Lindley distribution by mixing Poisson distribution with a two parameter Lindley distribution for modeling waiting and survival times data introduced by Shanker et al.¹³ Further, Shanker & Tekie¹⁴ obtained a new quasi Poisson-Lindley distribution by compounding Poisson distribution with a new quasi Lindley distribution introduced by Shanker & Amanuel.¹⁵

Recently Shanker¹⁶ has obtained Poisson-Sujatha distribution (PSD) having p.m.f. 

$P_0\left(x;\theta \right)=\frac{{\theta }^3}{{\theta }^2+\theta +2}\frac{x^2+\left(\theta +4\right)x+\left({\theta }^2+3\theta +4\right)}{{\left(\theta +1\right)}^{x+3}};x=0,1,2,3,\mathrm{...},\theta >0$       (1.4)

to model count data in different fields of knowledge. The PSD arises from the Poisson distribution when its parameter $\lambda$ follows Shanker R¹⁷ with probability density function (p.d.f.)

$g\left(\lambda ;\theta \right)=\frac{{\theta }^3}{{\theta }^2+\theta +2}\left(1+\lambda +{\lambda }^2\right)e^{-\theta \lambda };\lambda >0,\theta >0$                      (1.5)

Detailed discussion about its various properties, estimation of the parameter and applications for modeling lifetime data has been mentioned in Shanker¹⁷ and shown by Shanker¹⁷ that (1.5) is a better model than the exponential and Lindley³ distributions for modeling lifetime data.  Shanker & Hagos^18,19 has also obtained the size-biased and zero-truncated version of PSD and discussed their properties, estimation of parameter and applications in different fields of knowledge.   

In this paper, the nature of zero-truncated Poisson distribution (ZTPD), zero-truncated Poisson-Lindley distribution (ZTPLD) and zero-truncated Poisson-Sujatha distribution has been compared and studied using graphs for different values of their parameter. A simple method for obtaining the moments of ZTPSD has been suggested and the first two moments about origin and variance have been obtained. ZTPD, ZTPLD and ZTPSD have been fitted to a number of data -sets from demography and biological sciences to study their goodness of fit and superiority of one over the others.

Zero-truncated poisson distribution (ZTPD)

Using (1.1) and the p.m.f. of Poisson distribution, the p.m.f. of zero-truncated Poisson distribution (ZTPD) given by

$P_1\left(x;\theta \right)=\frac{{\theta }^x}{\left(e^{\theta }-1\right)x!};x=1,2,3,\mathrm{....},\theta >0$                                    (2.1.1)

was obtained independently by Plackett²⁰ and David & Johnson²¹ to model count data excluding zero counts. An extension of a truncated Poisson distribution and estimation in a truncated Poisson distribution when zeros and some ones are missing has been discussed by Cohen.^22,23 Tate & Goen²⁴ have discussed minimum variance unbiased estimation (MVUE) for the truncated Poisson distribution.

Zero-truncated poisson-lindley distribution (ZTPLD)

Using (1.1) and (1.2), the p.m.f. of zero-truncated Poisson- Lindley distribution (ZTPLD) given by

$P_2\left(x;\theta \right)=\frac{{\theta }^2}{{\theta }^2+3\theta +1}\frac{x+\theta +2}{{\left(\theta +1\right)}^x};x=1,2,3,\mathrm{....},\theta >0$              (2.2.1)

was obtained by Ghitany et al.²⁵ to model count data for the missing zeros.

Shanker et al.² have done extensive study on the comparison of ZTPD and ZTPLD with respect to their applications in data - sets excluding zero-counts and showed that in demography and biological sciences ZTPLD gives better fit than ZTPD while in social sciences ZTPD gives better fit than ZTPLD.

Zero-truncated poisson-sujatha distribution (ZTPSD)

Using (1.1) and (1.4), the p.m.f. of zero-truncated Poisson-Sujatha distribution (ZTPSD) can be obtained as

$P_3\left(x;\theta \right)=\frac{{\theta }^3}{{\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2}\frac{x^2+\left(\theta +4\right)x+\left({\theta }^2+3\theta +4\right)}{{\left(\theta +1\right)}^x};x=1,2,3,\mathrm{...},\theta >0$       (2.3.1)

The ZTPSD can also arise from the size-biased Poisson distribution (SBPD) with p.m.f.

$g\left(x|\lambda \right)=\frac{e^{-\lambda }{\lambda }^{x-1}}{\left(x-1\right)!};x=1,2,3,\mathrm{...},\lambda >0$                                              (2.3.2)

When its parameter $\lambda$ follows a distribution having p.d.f.

$h\left(\lambda ;\theta \right)=\frac{{\theta }^3}{{\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2}\left[{\left(\theta +1\right)}^2{\lambda }^2+\left(\theta +1\right)\left(\theta +3\right)\lambda +\left({\theta }^2+3\theta +4\right)\right]e^{-\theta \lambda }$ $,\lambda >0,\theta >0$     (2.3.3)

The p.m.f. of ZTPSD is thus can be obtained as

$P\left(x;\theta \right)={\displaystyle \underset{0}{\overset{\infty }{\int }}g\left(x|\lambda \right)}\cdot h\left(\lambda ;\theta \right)d\lambda$              $={\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{e^{-\lambda }{\lambda }^{x-1}}{\left(x-1\right)!}}\cdot \frac{{\theta }^3}{{\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2}\left[{\left(\theta +1\right)}^2{\lambda }^2+\left(\theta +1\right)\left(\theta +3\right)\lambda +\left({\theta }^2+3\theta +4\right)\right]e^{-\theta \lambda }d\lambda$  (2.3.4)                                                                                                                       

$=\frac{{\theta }^3}{\left({\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2\right)\left(x-1\right)!}{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\left(\theta +1\right)\lambda }}\cdot \left[\begin{array}{l}{\left(\theta +1\right)}^2{\lambda }^{x+2-1}+\left(\theta +1\right)\left(\theta +3\right){\lambda }^{x+1-1}\\ +\left({\theta }^2+3\theta +4\right){\lambda }^{x-1}\end{array}\right]d\lambda$

$=\frac{{\theta }^3}{\left({\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2\right)}\left[\frac{\left(x+1\right)x}{{\left(\theta +1\right)}^x}+\frac{\left(\theta +3\right)x}{{\left(\theta +1\right)}^x}+\frac{{\theta }^2+3\theta +4}{{\left(\theta +1\right)}^x}\right]$

$=\frac{{\theta }^3}{\left({\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2\right)}\frac{x^2+\left(\theta +4\right)x+\left({\theta }^2+3\theta +4\right)}{{\left(\theta +1\right)}^x}$

$x=1,2,3,\mathrm{...},\theta >0$

which is the p.m.f. of ZTPSD with parameter $\theta$ .

Shanker & Hagos¹⁹ have detailed study about its mathematical and statistical properties, estimation of parameter, and applications and showed that in many ways it has interesting advantage over ZTPLD and ZTPD.

To study the nature and behaviors of ZTPD, ZTPLD and ZTPSD for different values of their parameter, a number of graphs of their probability functions have been drawn and presented in Figure 1.

The $r$ ^th moment about origin of ZTPSD (2.3.1) can be obtained as

${\mu }_r{}^{\prime }=E\left[E\left(X^r|\lambda \right)\right]$       

$=\frac{{\theta }^3}{{\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2}{\displaystyle \underset{0}{\overset{\infty }{\int }}\left[{\displaystyle \sum _{x=1}^{\infty }{x}^r}\frac{e^{-\lambda }{\lambda }^{x-1}}{\left(x-1\right)!}\right]}\cdot \left[\begin{array}{l}{\left(\theta +1\right)}^2{\lambda }^2+\left(\theta +1\right)\left(\theta +3\right)\lambda \\ +\left({\theta }^2+3\theta +4\right)\end{array}\right]e^{-\theta \lambda }d\lambda$  (3.1)                                                                                   

Clearly the expression under the bracket in (3.1) is the $r$ ^th moment about origin of the SBPD. Taking $r=1$  in (3.1) and using the first moment about origin of the SBPD, the first moment about origin of the ZTPSD (2.3.1) can be obtained as

${\mu }_1{}^{\prime }=\frac{{\theta }^3}{{\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2}{\displaystyle \underset{0}{\overset{\infty }{\int }}\left(\lambda +1\right)}\left[\begin{array}{l}{\left(\theta +1\right)}^2{\lambda }^2+\left(\theta +1\right)\left(\theta +3\right)\lambda \\ +\left({\theta }^2+3\theta +4\right)\end{array}\right]e^{-\theta \lambda }d\lambda$

$=\frac{{\theta }^5+5{\theta }^4+15{\theta }^3+25{\theta }^2+20\theta +6}{\theta \left({\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2\right)}$                                                               (3.2)

Again taking $r=2$  in (3.1) and using the second moment about origin of the SBPD, the second moment about origin of the ZTPSD (2.3) can be obtained as

${\mu }_2{}^{\prime }=\frac{{\theta }^3}{{\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2}{\displaystyle \underset{0}{\overset{\infty }{\int }}\left({\lambda }^2+3\lambda +1\right)}\left[\begin{array}{l}{\left(\theta +1\right)}^2{\lambda }^2+\left(\theta +1\right)\left(\theta +3\right)\lambda \\ +\left({\theta }^2+3\theta +4\right)\end{array}\right]e^{-\theta \lambda }d\lambda$

$=\frac{{\theta }^6+7{\theta }^5+27{\theta }^4+73{\theta }^3+112{\theta }^2+84\theta +24}{{\theta }^2\left({\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2\right)}$                                              (3.3.3)

Similarly, taking $r=3\text{and}4$ in (3.1) and using the respective moments of SBPD, the third and the fourth moment about origin of ZTPSD can be obtained. The variance of ZTPSD (2.3. 1) are thus obtained as

${\mu }_2=\frac{{\theta }^9+10{\theta }^8+58{\theta }^7+210{\theta }^6+503{\theta }^5+760{\theta }^4+686{\theta }^3+352{\theta }^2+96\theta +12}{{\theta }^2{\left({\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2\right)}^2}$

$\gamma =\frac{{\sigma }^2}{\mu }=\frac{{\theta }^9+10{\theta }^8+58{\theta }^7+210{\theta }^6+503{\theta }^5+760{\theta }^4+686{\theta }^3+352{\theta }^2+96\theta +12}{\theta \left({\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2\right)\left({\theta }^5+5{\theta }^4+15{\theta }^3+25{\theta }^2+20\theta +6\right)}$

It can be easily verified that the ZTPSD is over dispersed $\left(\mu <{\sigma }^2\right)$ , equi-dispersed $\left(\mu ={\sigma }^2\right)$  and under dispersed $\left(\mu >{\sigma }^2\right)$  for $\theta <(=)>{\theta }^{\ast }=$ 1.548328 respectively. It is to noted that ZTPLD is over dispersed $\left(\mu <{\sigma }^2\right)$ , equi-dispersed $\left(\mu ={\sigma }^2\right)$ and under dispersed $\left(\mu >{\sigma }^2\right)$  for $\theta <(=)>{\theta }^{\ast }=$ 1.258627 respectively.

Estimation of parameter of ZTPD

Let $x_1,x_2,\mathrm{...},x_n$  be a random sample of size $n$  from the ZTPD (2.1.1). The maximum likelihood estimate (MLE) and method of moment estimate (MOME) of $\theta$  of ZTPD (2.1.1) is given by the solution of the following non linear equation

$e^{\theta }\left(\overline{x}-\theta \right)-\overline{x}=0$ , where $\overline{x}$ is the sample mean.

Estimation of parameter of ZTPLD

Maximum likelihood estimate (MLE):                                   

The maximum likelihood estimate $\widehat{\theta }$ of $\theta$  of ZTPLD is the solution of the following non-linear equation

$\frac{2n}{\theta }-\frac{n\left(2\theta +3\right)}{{\theta }^2+3\theta +1}-\frac{n\overline{x}}{\theta +1}+{\displaystyle \sum _{x=1}^k\frac{f_x}{x+\theta +2}}=0$                                     

Where $\overline{x}$ is the sample mean. This non-linear equation can be solved by any numerical iteration methods such as Newton-Raphson method, Bisection method, Regula-Falsi method etc. Ghitany et al.²⁵ showed that the MLE  $\widehat{\theta }$  of $\theta$ is consistent and asymptotically normal.

Method of moment estimate (MOME): Equating the population mean to the corresponding sample mean, the MOME $\tilde{\theta }$  of $\theta$  of ZTPLD (2.2.1) is the solution of the following cubic equation

$\left(\overline{x}-1\right){\theta }^3+\left(3\overline{x}-4\right){\theta }^2+\left(\overline{x}-5\right)\theta -2=0;\overline{x}>1$ , where $\overline{x}$ is the sample mean. Ghitany et al.²⁵ showed that the MOME $\tilde{\theta }$  of $\theta$ is consistent and asymptotically normal.

Maximum likelihood estimate (MLE):  The maximum likelihood estimate $\widehat{\theta }$ of $\theta$ of ZTPSD is the solution of the following non-linear equation

    ${\displaystyle \sum _{x=1}^k\frac{\left(x+2\theta +3\right)f_x}{x^2+\left(\theta +4\right)x+\left({\theta }^2+3\theta +4\right)}}-\frac{n\overline{x}}{\theta +1}-\frac{n\left({\theta }^4-10{\theta }^2-14\theta -6\right)}{\theta \left({\theta }^4+4{\theta }^3+10{\theta }^2+7\theta +2\right)}=0$           

Where $\overline{x}$ is the sample mean. This non-linear equation can be solved by any numerical iteration methods such as Newton-Raphson, Bisection method, Regula-Falsi method etc.

Method of moment estimate (MOME): Equating the population mean to the corresponding sample mean, the method of moment estimate (MOME) $\tilde{\theta }$ of $\theta$  of ZTPSD is the solution of the following non-linear equation

          $\left(1-\overline{x}\right){\theta }^5+\left(5-4\overline{x}\right){\theta }^4+\left(15-10\overline{x}\right){\theta }^3+\left(25-7\overline{x}\right){\theta }^2+\left(20-2\overline{x}\right)\theta +6=0$  Where $\overline{x}$ is the sample mean?

 Figure 1 Graph of probability mass functions of ZTPD, ZTPLD and ZTPSD for different values of their parameter.

In this section, an attempt has been made to test the suitability of ZTPD, ZTPLD and ZTPSD in describing the neonatal deaths as well as of infant and child deaths experienced by mothers. The data-sets considered here are the data of Sri Lanka and India. The data-sets of Meegama²⁶ have been used as the data of Sri Lanka whereas the data from the survey conducted by Lal²⁷ and the survey of Kadam Kuan, Patna, conducted in 1975 and referred to by Mishra²⁸ have been used as the data of India. Further, ZTPD, ZTPLD, and ZTPSD have also been fitted to data on migration. It is obvious from the fitting of ZTPD, ZTPLD and ZTPSD that ZTPSD and ZTPLD are competitive distributions for modeling data from demography (Table A1-A8).

In this paper, the nature and behavior of ZTPD, ZTPLD and ZTPSD have been studied by drawing different graphs of their probability functions for the different values of their parameter. A very simple and easy method for finding moments of ZTPSD has been suggested. An attempt has been made to study the goodness of fit of ZTPD, ZTPLD and ZTPSD to count data relating to demography and biological sciences and it has been observed that ZTPSD is a better model than the ZTPD and ZTPLD in almost all data-sets relating to mortality, migration and biological sciences.

None.

Author declares that there are no conflicts of interest.

1. Sankaran M. The discrete Poisson-Lindley distribution. Biometrics. 1970;26:145–149.
2. Shanker R, Hagos F. On Poisson-Lindley distribution and Its Applications to Biological Sciences. Biometrics and Biostatistics International Journal. 2015;2(4):1–5.
3. Lindley DV. Fiducial distributions and Bayes’ Theorem. Journal of the Royal Statistical Society. 1958;20(1):102–107.
4. Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and Its Applications. Mathematics Computation and Simulation. 2008a;78(4):493–506.
5. Shanker R, Hagos F, Sujatha S. On modeling of Lifetimes data using exponential and Lindley distributions. Biometrics and Biostatistics International Journal. 2015;2(5):1–9.
6. Shanker R, Hagos F, Sujatha S, et al. On Zero-truncation of Poisson and Poisson-Lindley distributins and Their Applications. Biometrics and Biostatistics International Journal. 2015;2(6):1–14
7. Ghitany ME, Al Mutairi DK. Estimation Methods for the discrete Poisson-Lindley distribution. Journal of Statistical Computation and Simulation. 2009;79(1):1–9.
8. Mishra A. A two-parameter Poisson-Lindley distribution. International Journal of Statistics and Systems. 2014;9(1):79–85.
9. Shanker R, Mishra A. A two-parameter Lindley distribution. Statistics in Transition new Series. 2013a;14(1):45–56.
10. Shanker R, Mishra A. A quasi Poisson-Lindley distribution. Accepted in Journal of Indian Statistical Association. 2015.
11. Shanker R, Mishra A. A quasi Lindley distribution. African journal of Mathematics and Computer Science Research. 2013b;6(4):64–71.
12. Shanker R, Sharma S, Shanker R. A Discrete two-Parameter Poisson Lindley Distribution. Journal of Ethiopian Statistical Association. 2012;21:15–22.
13. Shanker R, Sharma S, Shanker R. A two-parameter Lindley distribution for modeling waiting and survival times data. Applied Mathematics. 2013;4(2):363–368.
14. Shanker R, Tekie AL. A new quasi Poisson-Lindley distribution. International Journal of Statistics and Systems. 2013;9(1):87–94.
15. Shanker R, Amanuel AG. A new quasi Lindley distribution. International Journal of Statistics and Systems. 2013;6(4):143–156.
16. Shanker R. The discrete Poisson-Sujatha distribution. International Journal of Probability and Statistics. 2016b;5(1):1–9.
17. Shanker R. Sujatha distribution and Its Applications. Accepted for publication in “Statistics in Transition-new Series”. 2016a.
18. Shanker R, Hagos F. Size-Biased Poisson-Sujatha distribution with Applications, Communicated. 2016a.
19. Shanker R, Hagos F. Zero-truncated Poisson-Sujatha distribution with Applications. Communicated. 2016b.
20. Plackett RL. The truncated Poisson-distribution. Biometrics. 1953;9(4):485–488.
21. David FN, Johnson NL. The truncated Poisson. Biometrics. 1952;8:275–285.
22. Cohen AC. An extension of a truncated Poisson distribution. Biometrics. 1960a;16(3):446–450.
23. Cohen AC. Estimation in a truncated Poisson distribution when zeros and some ones are missing. Journal of American Statistical Association. 1960b;55:342–348.
24. Tate RF, Goen RL. MVUE for the truncated Poisson distribution. Annals of Mathematical Statistics. 1959;29:755–765.
25. Ghitany ME, Al Mutairi DK, Nadarajah S. Zero-truncated Poisson-Lindley distribution and its Applications. Mathematics and Computers in Simulation. 2008b;79(3):279–287.
26. Meegama SA. Socio-economic determinants of infant and child mortality in Sri Lanka, an analysis of postwar experience. International Statistical Institute. 1980;55.
27. Lal DN. Patna in 1955: A Demographic Sample Survey, Demographic Research Center, Department of Statistics, Patna University, Patna, India. 1955.
28. Mishra A. Generalizations of some discrete distributions. Patna University, Patna, India. 1979.
29. Singh SN, Yadav RC. Trends in rural out-migration at household level. Rural Demography. 1971;8:53–61.
30. Garman P. The European red mites in Connecticut apple orchards, Connecticut. Agri Exper Station Bull. 1953;252:103–125.
31. Student. On the error of counting with a haemacytometer. Biometrika. 1907;5(3):351–360.
32. Khursid AM. On size-biased Poisson distribution and Its use in zero-truncated case. Journal of the Korean Society for Industrial and Applied Mathematics. 2008;12(3):153–160.
33. Mathews JNS, Appleton DR. An application of the truncated Poisson distribution to Immunogold assay. Biometrics. 1953;49(2):617–621.
34. Keith LB, Meslow EC. Trap response by snowshoe hares. The Journal of Wildlife Management. 1968;32(4):795–801.
35. Finney DJ, Varley GC. An example of the truncated Poisson distribution. Biometrics. 1955;11(3):387–394.