The Poisson-weighted Sujatha Distribution with Properties and Applications

Received: 02/Sept/2018, Accepted 11/Oct/2018, Online 31/Oct/2018 AbstractThe Poisson distribution is an important discrete distribution for modeling count data having equi-dispersion. In this paper, a new discrete distribution for modeling count data having over-dispersion, namely, the Poisson-weighted Sujatha distribution which includes Poisson-Sujatha distribution has been proposed by compounding Poisson distribution with a twoparameter weighted Sujatha distribution. Its statistical properties including moments, coefficient of variation, skewness, kurtosis, index of dispersion, unimodality and increasing hazard rate have been discussed. Maximum likelihood estimation has been explained for estimating its parameters. Applications of the distribution have been discussed with some count datasets and its goodness of fit has been compared with other discrete distributions having over-dispersion.

It has been observed by Karlis and Xekalaki [2] that there are naturally situations where a good fit is not obtainable with a particular mixed Poisson distribution in case of over-dispersed count data. This shows that there is a requirement for new mixed Poisson distribution which gives better fit as compared with the existing mixed Poisson distributions.
Shanker [15] introduced Poisson-Sujatha distribution (PSD) defined by its probability mass function (pmf)         ; 0,1, 2,..., 0 2 1 Statistical properties including shapes of pmf for varying values of parameter, coefficient of variation, skewness, kurtosis, index of dispersion, unimodality, increasing hazard rate have been discussed by Shanker [15]. Further, the estimation of parameter using both the method of moment and the method of maximum likelihood along with applications of PSD has been discussed by Shanker [15]. Note that PSD arises from the Poisson distribution where the mixing distribution is the Sujatha distribution introduced by Shanker [ Shanker [16] has discussed several important statistical and mathematical properties of Sujatha distribution including moments and moments based measures, hazard rate function, mean residual life function, mean deviations from the mean and the median, stochastic ordering, Bonferroni and Lorenz curves and stress-strength reliability, are some among others. Further, Shanker [16] has discussed the method of moment and the method of maximum likelihood estimation for estimating the parameter of Sujatha distribution. The applications and goodness of fit of Sujatha distribution have also been discussed by Shanker [16] with some real lifetime datasets from biomedical sciences and engineering. Shanker [15] has shown that PSD is a better model than the Poisson-Lindley distribution (PLD) introduced by Sankaran [ ; ,  (1.2). This distribution has been found to be a better model than the one parameter Sujatha distribution for analyzing and modeling lifetime data from engineering. Shanker and Shukla [17] have derived and discussed various statistical properties of WSD including moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, some among others. Further, Shanker and Shukla [17] have discussed the method of maximum likelihood for estimating the parameter of WSD.
; , Its structural properties including moments, hazard rate function, mean residual life function, estimation of parameters and applications for modeling survival time data has been discussed by Ghitany et al [14]. The corresponding cumulative distribution function (cdf) of WLD (1.5) is given by is the upper incomplete gamma function. It can be easily shown that at 1   , WLD reduces to Lindley [6] distribution.
Shanker et al [18] discussed various moments based properties including coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of WLD and its applications to model lifetime data from biomedical sciences and engineering. Shanker et al [19] have proposed a three-parameter weighted Lindley distribution (TPWLD) which includes a twoparameter weighted Lindley distribution and one parameter Lindley distribution as particular cases and discussed its various structural properties, estimation of parameters and applications for modeling lifetime data from engineering and biomedical sciences.
Assuming that the parameter  of the Poisson distribution follows WLD (1.5), Abd El-Monsef and Sohsah [13] proposed Poisson-weighted Lindley distribution (P-WLD) defined by its pmf It can be easily verified that PLD (1.3) is a particular case of P-WLD for 1   .
Since WSD is a better model than Sujatha, Lindley and weighted Lindley distributions for modeling lifetime data and PSD is a better model than PLD for modeling count data, it is expected and hoped that a Poisson mixture of WSD will provide a better model than Poisson, PLD, PSD and P-WLD for modeling count data. Keeping these points in mind, a Poisson mixture of WSD has been introduced and studied. Its various statistical properties based on moments, increasing hazard rate and estimation of parameters using the method of maximum likelihood along with applications have been discussed.

II. THE POISSON-WEIGHTED SUJATHA DISTRIBUTION
Assuming that the parameter  of the Poisson distribution follows WSD (1.5), the Poisson mixture of WSD can be obtained as

A. Factorial Moments
Using (2.1), the r th factorial moment about origin,   r   of the P-WSD (2.2) can be obtained as

B.Raw Moments (Moments about Origin)
Using the relationship between raw moments and the factorial moments about origin, the raw moments of P-WSD are obtained as

C. Central Moments (Moments about Mean)
Now, using the relationship    

V. GOODNESS OF FIT
In this section the applications of the P-WSD has been discussed with two count datasets from biological sciences. The dataset in table 1 is the data regarding the number of European red mites on apple leaves, available in Bliss [21]. The dataset in table 2 figure 3. It is obvious that P-WSD gives much closer fit over other mixed Poisson distributions.

VI.CONCLUDING REMARKS
The Poisson-weighted Sujatha distribution (P-WSD) which includes Poisson-Sujatha distribution (PSD) as particular case has been proposed by compounding Poisson distribution with a two-parameter weighted Sujatha distribution. Its statistical properties including moments, coefficient of variation, skewness, kurtosis, index of dispersion, unimodality and increasing © 2018, IJSRMSS All Rights Reserved 242 hazard rate have been discussed. Maximum likelihood estimation has been explained for estimating its parameters. Applications of the distribution have been discussed with two count datasets from biological sciences and its goodness of fit has been compared with other discrete distributions having over-dispersion. Since the fit by P-WSD has been found quite satisfactory over, PD, PLD, PSD, and P-WLD, P-WSD can be considered an important Poisson mixed distribution in distribution theory.