On Stuttering Hyper-Poisson Distribution and its Properties

Here we develop an order k version of the hyper-Poisson distribution and study some of its properties by deriving its probability mass function, mean, variance and recursion formulae for probabilities, raw moments and factorial moments. The estimation of the parameters of this class of distributions by the method of mixed moments and method of maximum likelihood is attempted and it is demonstrated with the help of a real data set that this order k version of the hyper-Poisson distribution fits the situations better than the existing model.


Introduction
has obtained a two-parameter generalization of the Poisson distribution, namely the hyper-Poisson distribution, which they defined as in the following. A non-negative integer valued random variable X is said to follow the hyper-Poisson distribution (HPD) if its probability mass function (p.m.f.) has the following form. t Gt        (3) When =1  , the HPD reduces to the Poisson distribution and when  is a positive integer, the distribution is known as the displaced Poisson distribution of Staff (1964). Bardwell and Crow (1964) termed the distribution as sub-Poisson when <1  and super-Poisson when >1  . Various methods of estimation of the parameters of the distribution were discussed in Bardwell and Crow (1964) and Crow and Bardwell (1965). Some queuing theory associated with hyper-Poisson arrivals has been worked out by Nisida (1962). Roohi and Ahmad (2003a) attempted estimation of the parameters of the HPD using negative moments. Roohi and Ahmad (2003b) derived expressions for ascending factorial moments and further obtained certain recurrence relations for negative moments and ascending factorial moments of the HPD. Kemp (2002) developed a q-analogue of the distribution and Ahmad (2007) introduced and studied Conway-Maxwell hyper-Poisson distribution. Kumar andNair (2011, 2012) developed extended versions of the HPD.
In this article, we obtain an order k version of the HPD and call it as "the stuttering hyper-Poisson distribution" or in short "the SHPD". In section 2 we establish that the SHPD possess a stopped sum structure. In section 3, we present some of its properties by deriving expressions for its probability mass function, mean and variance. We also obtain certain recursion formulae for probabilities, raw moments and factorial moments in section 3. Moreover, the estimation of the parameters of the SHPD has been discussed in section 4 by the method of mixed moments using first observed frequency and the method of maximum likelihood. Further, a generalized likelihood ratio test is considered for testing the significance of the additional parameter in section 5. All these estimation and testing procedures are illustrated with the help of a real data and presented in respective sections. In section 6. we have conducted a simulation study for examining the performance of the estimators of the parameters. Since the SHPD possess a stopped sum structure, they may be useful for modeling real world phenomena arising from various fields of research such as actuarial science, biological sciences, operations research and physical sciences.  We define a distribution with p.g.f. (5), as the stuttering hyper-Poisson distribution or in short the SHPD . When  is a positive integer, the distribution is known as stuttering displaced Poisson distribution and when =1  we get the stuttering Poisson distribution of Galliher et.al. (1959). The stuttering Poisson distribution was further studied by Aki (1985), Philippou (1988), Moothathu and Kumar (1995) and Kumar (2009

Properties
Let W be a random variable distributed as the SHPD with p.g.f. (5). Here, first we obtain an expression for the p.m.f. of the SHPD through the following result. ISBN-1391-4987 . On equating the coefficients of x z on both sides of (7) we get the p.m.f. of the SHPD as given in (6).
Next we derive expressions for the mean and variance of the SHPD through the following result.  (9) where for 1,2,...
The proof follows from the fact that Here after we denote the p.m.f.
x h of the SHPD by (1, ) x h  . Thus, from (5) we have the following.
Now we have the following results.

Result 3.3
The following is a simple recursion formula for the probabilities ( (12) in which 1  is as defined in (10).
Proof. Differentiate (11) with respect to z , to get On replacing  by 1   in (11) we obtain the following.
Relations (13) and (14) together lead to the following relationship: Now, on equating coefficients of x z on both sides of (15) we get (12).

Result 3.4
The following is a recursion formula for factorial moments (16) where () = ( 1)( 2)... On differentiating (17) with respect to t to obtain By using (17) with  replaced by 1   we get the following from (18). On

Estimation
In this section we consider the estimation of the parameters of the SHPD by the method of mixed moments and the method of maximum likelihood.

Method of mixed moments
In this method, the parameters  , 1  , 2  , ..., k  (for a fixed value of k ) of the SHPD are estimated by using the first k sample raw moments and the first observed ISBN-1391-4987 ©IASSL frequency of the distribution. The first k raw moments of the SHPD are equated to the corresponding sample raw moments (say 1  , 2  ,... , k  ) to get ( (25) in which 0 h is the observed frequency of the distribution corresponding to the observation zero and N is the total observed frequency.

Method of maximum likelihood
It is difficult to obtain explicit expressions for the parameters of the SHPD in both mixed moments and maximum likelihood methods of estimation. The likelihood equations (29) and (30) do not always have a solution because the SHPD is not a regular model. Therefore, when likeihood equation does not have a solution, the maximum of the likelihood function is attained at the border of the domain of parameters. As such we obtained the second order partial derivatives of the function (1, ) x h  with respect to the parameters  , 1  , 2  ,…, k  and we observed, with the help of MATHCAD software, that these equations are negative for all >0  and >0  Table 1 and Table 2 for 1,2 k  and 3 by using MATHCAD software.
From the calculated chi-square values, P -values, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) in each case, it can be seen that the SHPD with =3 k gives a better fit to the given data than the standard hyper-Poisson distribution ( = 1) k as well as the distribution with =2 k . x logL x  and the test statistic for the SHPD and presented in Table 3.  Since the critical value for the test with significance level equal to 0.05 and degrees of freedom one is 3.84, The null hypothesis is rejected in this case.

Simulation
Three data sets were simulated three data sets for the following three sets of parameters. Note that the first two data sets simulate under-dispersed distributions, and the third one simulates an over-dispersed distribution. Bias and standard error for each estimated parameter are then obatined and given below: From the Table 4 it can be observed that both bias and standard errors are in decreasing order as the sample size increases.