Zero-Truncated Poisson-Garima Distribution and its Applications

This distribution has been extensively studied by Shanker [2] and it has been shown that (1.3) provides a better model for behavioral science data than many one parameter exponential, Lindley distribution introduced by Lindley[3], Shanker, Akash, Aradhana and Sujatha distributions introduced by [3-7]. In the present paper an attempt has been made to obtain a zero-truncated Poisson-Garima distribution (ZTPGD) by taking the zero-truncated version of Poisson-Garima distribution (PGD) of shanker [1]. Its moments and moments based properties including coefficients of variation, skewness and kurtosis, and index of dispersion of ZTPGD have been obtained and discussed graphically. For estimating the parameter the method of moment and the method of maximum likelihood estimation have been discussed. Goodness of fit of ZTPGD has been discussed with two real data sets and the fit has been compared with that of zero-truncated Poisson distribution (ZTPD) and zero-truncated Poisson-Lindley distribution (ZTPLD) [8].

In probability theory, zero-truncated distributions are certain discrete distributions whose support is the set of positive integers. When the data to be modeled originate from a mechanism which generates data that structurally excludes zero counts, zero-truncated distribution is the appropriate choice. This distribution has been extensively studied by Shanker [2] and it has been shown that (1.3) provides a better model for behavioral science data than many one parameter exponential, Lindley distribution introduced by Lindley [3], Shanker, Akash, Aradhana and Sujatha distributions introduced by [3][4][5][6][7].
In the present paper an attempt has been made to obtain a zero-truncated Poisson-Garima distribution (ZTPGD) by taking the zero-truncated version of Poisson-Garima distribution (PGD) of shanker [1]. Its moments and moments based properties including coefficients of variation, skewness and kurtosis, and index of dispersion of ZTPGD have been obtained and discussed graphically. For estimating the parameter the method of moment and the method of maximum likelihood estimation have been discussed. Goodness of fit of ZTPGD has been discussed with two real data sets and the fit has been compared with that of zero-truncated Poisson distribution (ZTPD) and zero-truncated Poisson-Lindley distribution (ZTPLD) [8].

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when its parameter follows a continuous distribution having p.d.f.
The pmf of zero-truncated Poisson-Lindley distribution (ZTPLD) obtained by Ghitany et al. [9] is given by Recall that ZTPLD has been obtained by zero-truncating the discrete Poisson-Lindley distribution suggested by Sankaran [10] and the discrete Poisson-Lindley distribution is the Poisson mixture of Lindley distribution introduced by Lindley [3]. Shanker & Hagos [11] have detailed study on applications of Poisson-Lindley distribution for biological sciences. Shanker et al. [12] have detailed study on modeling of lifetime data using both exponential and Lindley distributions and concluded that both exponential and Lindley distributions compete each other.

Moments and Related Measures
Using (2.4), the th factorial moment about origin of ZTPGD (2.1) can be obtained as Substituting in (3.1), the first four factorial moments can be obtained and using the relationship between moments about origin and factorial moments about origin, the first four moments about origin of ZTPGD can be obtained as   Table 1.
is a decreasing function of , is log-concave. Therefore, ZTPGD is unimodal, has increasing failure rate (IFR), and hence increasing failure rate average (IFRA). It is new better than used (NBU), new better than used in expectation (NBUE), and has decreasing mean residual life (DMRL). Detailed discussions about the definitions of these aging concepts are available in Barlow & Proschan [13].

Generating Function
Probability Generating Function: The probability generating function of the ZTPGD (2.1) is obtained as

Method of Moment Estimate (MOME)
Let be a random sample of size from the ZTPGD (2.1). Equating the population mean to the corresponding sample mean, MOME of is the solution of the following non-linear equation

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where x is the sample mean.

Maximum Likelihood Estimate (MLE)
Let 1 2 , ,..., n x x x be a random sample of size from the ZTPGD

Applications
The ZTPGD has been fitted to a number of data -sets to test its goodness of fit over ZTPD and ZTPLD. The maximum likelihood estimate (MLE) has been used to fit the ZTPGD. Two examples of observed data-sets, for which the ZTPD, ZTPLD and ZTPGD has been fitted, are presented. The first data-set in Table 2 is the number of European red mites on apple leaves, reported by Garman [14] and the second data-set in Table 3 is the animal abundance data of Keith & Meslow [15] regarding the distribution of snowshoe hares captured over 7 days (Figure 3).

Concluding Remarks
A zero-truncated Poisson-Garima distribution (ZTPGD) has been introduced by taking the zero-truncated version of Poisson-Garima distribution (PGD) of Shanker [1]. To find the moments of the distribution ZTPGD has also been obtained by taking a size-biased mixture of an assumed continuous distribution. Its raw moments and central moments and moments based properties including coefficients of variation, skewness and kurtosis, and index of dispersion have been obtained and discussed graphically. Both the method of moment and the method of maximum likelihood estimation have been discussed for estimating the parameter. Finally two examples of real data sets have been presented to test the goodness of fit of ZTPGD and the fit shows quite satisfactory fit over ZTPD and ZTPL