ZERO-TRUNCATED DISCRETE TWO-PARAMETER POISSON-LINDLEY DISTRIBUTION WITH APPLICATIONS

A zero-truncated discrete two-parameter Poisson-Lindley distribution (ZTDTPPLD), which includes zerotruncated Poisson-Lindley distribution (ZTPLD) as a particular case, has been introduced. The proposed distribution has been obtained by compounding size-biased Poisson distribution (SBPD) with a continuous distribution. Its raw moments and central moments have been given. The coefficients of variation, skewness, kurtosis, and index of dispersion have been obtained and their nature and behavior have been studied graphically. Maximum likelihood estimation (MLE) has been discussed for estimating its parameters. The goodness of fit of ZTDTPPLD has been discussed with some data sets and the fit shows satisfactory over zero – truncated Poisson distribution (ZTPD) and ZTPLD.


INTRODUCTION
In probability theory, zero-truncated distributions are certain discrete distributions having support the set of positive integers.Zero-truncated distributions are suitable models for modeling data when the data to be modeled originate from a mechanism which generates data excluding zero counts.; 0,1,2,..., 0, 1 It can be easily verified that at 1   , DTPPLD (1.2) reduces to the one parameter Poisson-Lindley distribution (PLD) introduced by Sankaran (1970) having pmf       2 2 3 2 ; ; 0,1, 2,..., 0 1 (1.3) Shanker et al. (2012) have studies the mathematical and statistical properties, estimation of parameters of DTPPLD and its applications to model count data.It should be noted that PLD is also a Poisson mixture of Lindley distribution, introduced by Lindley (1958).Shanker and Hagos (2015) have discussed the applications of PLD for modeling data from biological sciences.The DTPPLD is a Poisson mixture of a twoparameter Lindley distribution (TPLD) of Shanker et al. (2013) having probability density function (pdf) In this paper, a ZTDTPPLD, of which zerotruncated Poisson-Lindley distribution (ZTPLD) is a particular case, has been obtained by compounding size-biased Poisson distribution (SBPD) with a continuous distribution.Its raw moments and central moments have been obtained and thus the expressions for coefficient of variation, skewness, kurtosis, and index of dispersion have been obtained and their nature and behavior have been discussed graphically.Maximum likelihood estimation has been discussed for estimating the parameters of ZTDTPPLD.The goodness of fit of ZTDTPPLD has also been discussed with some data sets and its fit has been compared with zero -truncated Poisson distribution (ZTPD) and zerotruncated Poisson-Lindley distribution (ZTPLD).Using (1.1) and (1.2), the pmf of zero-truncated discrete two-parameter Poisson-Lindley distribution (ZTDTPPLD) can be obtained as

ZERO-TRUNCATED DISCRETE TWO-PARAMETER POISSON-LINDLEY DISTRIBUTION
To study the nature and behavior of ZTDTPPLD for varying values of parameters  and  , a number of graphs of the pmf of ZTDTPPLD have been drawn and presented in the figure 1.

Fig.1. Graph of the probability mass function of ZTDTPPLD for varying values of parameters α and θ.
The ZTDTPPLD (2.1) can also be obtained from size-biased Poisson distribution (SBPD) having pmf when the parameter  of SBPD follows a continuous distribution having pdf (2.5) Thus, the pmf of ZTDTPPLD can be obtained as which is the pmf of ZTDTPPLD with parameter  and  as given in (2.1).

MOMENTS OF ZTDTPPLD
The r th factorial moment about origin of ZTDTPPLD (2.1) can be obtained as Using gamma integral and a little algebraic simplification, we get the expression for the r th factorial moment about origin of ZTDTPPLD (2.1) as Substituting 1, 2,3, and 4 r  in equation (3.1), the first four factorial moments about origin can be obtained and using the relationship between moments about origin and factorial moments about origin, the first four moments about origin of ZTDTPPLD (2.1) are obtained as Again using the relationship between moments about origin and moments about mean, the moments about mean of ZTDTPPLD (2.1) are obtained as The nature of coefficient of variation, coefficient of skewness, coefficient of kurtosis, and index of dispersion of ZTDTPPLD (2.1) are shown graphically in figure 2.

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS
Let   12 , ,..., n x x x be a random sample of size n from the ZTDTPPLD (2.1) and let x f be the observed frequency in the sample corresponding to ( 1, 2,3,..., ) , where k is the largest observed value having non-zero frequency.The likelihood function L of the ZTDTPPLD (2.1) is given by The log likelihood function is thus obtained as where x is the sample mean.
These two log likelihood equations do not seem to be solved directly.However, the Fisher's scoring method can be applied to solve these equations.We have where 0  and 0  are the initial values of  and  , respectively.These equations are solved iteratively till sufficiently close values of  and  are obtained.In this paper R software has been used to estimate parameters of the ZTDTPPLD.

GOODNESS OF FIT
In this section, we present the goodness of fit of ZTDTPPLD, ZTPD and ZTPLD for four count data sets.The first data set is due to Finney and Varley (1955) who gave counts of number of flower having number of fly eggs.The second data set is due to Singh and Yadav (1971) regarding the number of households having at least one migrant from households according to the number of migrants.The third data set is regarding the number of counts of sites with particles from Immunogold data, reported by Mathews and Appleton (1993).The fourth data set is regarding the number of snowshoe hares counts captured over 7 days, reported by Keith and Meslow (1968).

Table 4 : The number of snowshoe hares counts captured over 7 days, reported by Keith and Meslow (1968) Number of times hares caught Observed frequency Expected Frequency
It is obvious from the goodness of fit of ZTDTPPLD, ZTPD, and ZTPLD that ZTDTPPLD gives better fit in tables 1, 2 and 3, while in table 4 ZTPLD gives better fit.The nature of the probability mass functions of the fitted distributions, ZTDTPPLD, ZTPLD, and ZTPD for four data sets has been shown graphically in the figure3.