1. Introduction

Poisson distribution is the common distribution for modeling count data in statistics for equi-dispersed (mean equal to variance) data. But the equality of the mean and the variance of Poisson distribution makes it unsuitable for modeling count data which are under-dispersed (mean greater than variance) or over-dispersed (mean less than variance). In recent years, several researchers have proposed Poisson mixture of lifetime distributions which are useful for over-dispersed or under-dispersed. Some one parameter over-dispersed Poisson mixed distributions are Poisson-Lindley distribution (PLD), a Poisson mixture of Lindley distribution of Lindley (1958) proposed by Sankaran (1970), Poisson-Sujatha distribution (PSD), a Poisson mixture of Sujatha distribution of Shanker (2016 a) introduced by Shanker (2016 b), among others.

The probability density function (pdf) of Sujatha distribution having scale parameter and introduced by Shanker (2016a) is

(1.1)

Statistical properties including shapes of the density, moments and moments based measures, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviation, stress-strength reliability, along with the estimation of parameter and applications for modeling lifetime data from biomedical science and engineering of Sujatha distribution are available in Shanker (2016 a). Kaliraja and Perarasan (2019) studied a stochastic model on the generalization of Sujatha distribution for the effects of two types of exercise on plasma growth hormone.

Shanker (2016 b) obtained Poisson-Sujatha distribution (PSD) by compounding Poisson distribution with Sujatha distribution. The PSD is defined by its probability mass function (pmf)

(1.2)

Statistical properties including shapes of pmf, moments and moments based measures, over-dispersion, unimodality and increasing hazard rate, estimation of parameters and applications o model over-dispersed data have been discussed by Shanker (2016 b). Wesley et al (2018) proposed a zero-modified Poisson-Sujatha distribution to model over-dispersed count data and discussed its several important properties and applications.

Shanker et al (2017) have introduced a generalization of Sujatha distribution (AGSD) having pdf and cdf given by

(1.3)

Shanker et al (2017) have discussed important statistical properties including shapes of the density, moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, stress-strength reliability, along with estimation of parameters using maximum likelihood estimation and applications of AGSD for modeling lifetime data from engineering and medical sciences. It can be easily verified that at , the pdf of AGSD reduces to the corresponding pdf of Sujatha distribution. Also, at , the pdf of AGSD reduces to Lindley distribution introduced by Lindley (1958).

Shanker and Shukla (2019) introduced a generalization of Poisson-Sujatha distribution (AGPSD) by compounding Poisson distribution with AGSD (1.3) and obtained the pmf in the form

(1.3)

Shanker and Shukla (2020) proposed a two-parameter Poisson-Sujatha distribution (TPPSD) defined by its pmf

(1.4)

It should be noted that TPPSD is a Poisson mixture of a two-parameter Sujatha distribution (TPSD) introduced by Mussie and Shanker (2018) and defined by its pdf

(1.5)

Recently, Mussie and Shanker (2019) proposed another two-parameter Sujatha distribution (ATPSD) defined by its pdf

(1.6)

where is a scale parameter and is is a shape parameter. It can be easily verified that (1.3) reduces to exponential distribution and Sujatha distribution for and respectively.

The main motivation for proposing ATPPSD are (i) Sujatha distribution is a better model than both exponential and Lindley distribution for modeling lifetime data, and PSD being a Poisson mixture of Sujatha distribution gives better fit than both Poisson and Poisson-Lindley distribution (PLD), (ii) TPSD gives much better fit than exponential, Lindley and Sujatha distribution, TPPSD being a Poisson mixture of TPSD provides better fit over PLD, PSD and other discrete distributions, and (iii) have a comparative study of ATPPSD with other two-parameter generalizations of Poisson-Sujatha distributions including TPPSD and AGPSD.

Keeping these points in mind, another two-parameter Poisson-Sujatha distribution (ATPPSD), a Poisson mixture of ATPSD has been proposed and its moments and moments based measures have been obtained and their behaviors have been studied. Maximum likelihood estimation of ATPPSD has been discussed for the estimation of its parameters. Its applications have been discussed with five examples of observed count datasets from various fields of knowledge.

2. Another Two-Parameter Poisson-Sujatha Distribution

A random variable is said to follow another two-parameter Poisson-Sujatha distribution (ATPPSD) if

and .

That is,

, and

The pmf of unconditional random variable can be obtained as

(2.1)

(2.2)

We would call this another two-parameter Poisson-Sujatha distribution (ATPPSD) because for , it reduces to one parameter PSD given in (1.2). Also at , it reduces to geometric distribution.

It can be easily shown that ATPPSD is unimodal and has increasing hazard rate. Since

is decreasing function in is log-concave. Now using the results of relationship between log-concavity, unimodality and increasing hazard rate (IHR) of discrete distributions available in Grandell (1997), it can concluded that ATPPSD has an increasing hazard rate and unimodal.

The behavior of the pmf of ATPPSD for varying values of parameters and are shown in figure 1.

Figure 1. Behaviour of pmf of ATPPSD for varying values of parameters and

3. Moments Based Measures

The factorial moment about origin of ATPPSD can be obtained as

, where . Using (2.1), the factorial moment about origin of ATPPSD can be obtained as

Taking within the bracket, we get

After some tedious algebraic simplification, a general expression for the factorial moment about origin of ATPPSD can be expressed as

(3.1)

It can be easily verified that at and , the expression (3.1) reduces to the corresponding expression of geometric distribution and PSD. Substituting in (3.1), the first four factorial moments about origin of ATPPSD can be obtained as

Now using the relationship between factorial moments about origin and moments about origin, the first four moment about origin of the ATPPSD are obtained as

Using the relationship between moments about mean and the moments about origin, the moments about mean of ATPPSD are obtained as

The coefficient of variation , coefficient of Skewness , coefficient of Kurtosis , and index of dispersion of ATPPSD are thus given by

It can be easily verified that at and expressions of these statistical constants of ATPPSD reduce to the corresponding expressions for geometric distribution and PSD.

The behaviors of coefficient of variation (C.V), coefficient of skewness (C.S), coefficient of kurtosis (C.K) and index of dispersion (I.D) of ATPPSD for varying values of parameters and have been explained through graphs and presented in figure 2.

Figure 2. Behaviors of coefficient of variation (C.V), coefficient of skewness (C.S), coefficient of kurtosis (C.K) and index of dispersion (I.D) of ATPPSD for varying values of parameters and

4. Maximum Likelihood Estimation of Parameters

Suppose be a random sample of size from ATPPSD and be the observed frequency in the sample corresponding to such that , where is the largest observed value having non-zero frequency. The likelihood function of ATPPSD is given by

The log - likelihood function is thus obtained as

The maximum likelihood estimates of parameters of ATPPSD is the solutions of the following log - likelihood equations

where is the sample mean. These two log likelihood equations do not seem to be solved directly because they do not have closed forms. Therefore, to find the maximum likelihood estimates of parameters, an iterative method such as Fisher Scoring method, Bisection method, Regula Falsi method or Newton-Raphson method can be used. In this paper Newton-Raphson method has been used using R-software.

5. Goodness of Fit

In this section five count datasets which are over-dispersed have been considered for testing the goodness of fit of ATPPSD and the fit has been compared with PLD, PSD, AGPSD and TPPSD. The goodness of fit of all these distributions is based on maximum likelihood estimation. The first dataset is regarding the number of European corn-borer available in Gosset (1908) , the second dataset is regarding the number of European corn-borer available in Mc Guire et al (1957), the third dataset is regarding the accidents to 647 women working on high explosive shells in 5 weeks, available in Sankaran (1970), the fourth dataset is regarding the number of European red mites on Apple leaves, available in Bliss (1953) and the fifth dataset is regarding the observed number of households according to the number of male migrants aged 15 years and above, available in Shukla and Yadav (2006). The maximum likelihood estimates, chi-squares, value of and Akaike information criterion (AIC) for the considered distributions for the given datasets have been computed and presented in the respective table. The AIC has been calculated using the formula , where is the number of parameters involved in the distribution. In table 1, AGPSD and ATPPSD give almost the same fit. In table 2, AGPSD gives the best fit, whereas TPPSD and ATPPSD gives the second best fit. In table 3, PSD gives the best fit whereas TPPSD and ATPPSD gives the second best fit. TPPSD and ATPPSD gives the same fit in table 4 and 5, respectively. Therefore, we can say that ATPPSD is competing well with AGPSD, and TPPSD for count datasets.

Table 1. Observed and expected number of European corn-borer available in Gosset (1908)

Table 2. Observed and expected number of European corn-borer available in Mc Guire et al (1957)

Table 3. Accidents to 647 women working on high explosive shells in 5 weeks, available in Sankaran (1970)

Table 4. Observed and Expected number of European red mites on Apple leaves, available in Bliss (1953)

Table 5. Observed and Expected number of households according to the number of male migrants aged 15 years and above, available in Shukla and Yadav (2006)

6. Concluding Remarks

In this paper, another two-parameter Poisson Sujatha distribution (ATPPSD) which includes geometric distribution and Poisson-Sujatha distribution (PSD) proposed by Shanker as a special case has been proposed.. Its unimodality, increasing hazard rate, moments and moments based measures including coefficients of variation, skewness, kurtosis and index of dispersion has been obtained and their behaviors have been explained graphically for varying values of parameters. The method of maximum likelihood estimation has been discussed. The applications of the proposed distribution has been explained through two examples of count data from ecology and the goodness of fit of the distribution has been found quite satisfactory over PLD, PSD, AGPSD, TPPSD.

ACKNOWLEDGEMENTS

Authors are grateful to the editor in chief of the journal and the anonymous reviewer for useful comments.