Bivariate Poisson-Lindley Distribution with Application

Corresponding Author: Noriszura Ismail School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia Email: ni@ukm.edu.my Abstract: This study applies a Bivariate Poisson-Lindley (BPL) distribution for modeling dependent and over-dispersed count data. The advantage of using this form of BPL distribution is that the correlation coefficient can be positive, zero or negative, depending on the multiplicative factor parameter. Several properties such as mean, variance and correlation coefficient of the BPL distribution are discussed. A numerical example is given and the BPL distribution is compared to Bivariate Poisson (BP) and Bivariate Negative Binomial (BNB) distributions which also allow the correlation coefficient to be positive, zero or negative. The results show that BPL distribution provides the smallest Akaike Information Criterion (AIC), indicating that the distribution can be used as an alternative for fitting dependent and over-dispersed count data, with either negative or positive correlation.

Based on literatures, the mixture approaches have been used to derive new families of bivariate distribution. The Bivariate Negative Binomial (BNB), Bivariate Poisson-Lognormal (BPLN), Bivariate Poisson-Inverse Gaussian (BPIG) and bivariate Poisson-Lindley (BPL) distributions are several examples of classes of mixed distribution which are extended from univariate case. For further literatures, BNB distribution was studied in Marshall and Olkin (1990) and applied in Karlis and Ntzoufras (2003), tests for overdispersion and independence in BNB model were discussed in Cheon et al., 2009), BPIG distribution was derived by Stein et al. (1987), BPL distribution was proposed by Gomez-Deniz et al. (2012) and Bivariate Poisson-Weighted Exponential (BPWE) was proposed in Zamani et al. (2014b).
Besides mixture approach, several bivariate discrete distributions have been defined using the method of trivariate reduction (Kocherlakota and Kocherlakota, 1999;Johnson et al., 1997). The BP distribution from the trivariate reduction has been used for modeling correlated bivariate count data and several applications can be found in (Holgate, 1964;Paul and Ho, 1989). Besides BP distribution, the Bivariate Generalized Poisson (BGP) distribution from the trivariate reduction has been defined and studied in Famoye and Consul (1995), where the distribution can be used for modeling correlated and under-or overdispersed bivariate count data.
In this study, we apply the BPL distribution which was derived from the product of two PL marginals with a multiplicative factor parameter. This BPL distribution can be used for bivariate count data with positive, zero or negative correlation coefficient. The rest of this study is organized as follows. Section 2 provides the univariate version of PL distribution. Several properties of the BPL distribution, such as mean, variance and correlation coefficient, are discussed in section 3. Section 4 discusses parameter estimation for the BPL and section 5 provides several tests for testing independence. Numerical illustration is provided in section 6, where BPL distribution is fitted to the bivariate flight aborts count data. The BPL distribution is compared to BP (Lakshminarayana et al., 1999) and BNB (Famoye, 2010) distributions which also allow positive, zero or negative correlation.

Univariate Poisson-Lindley (PL) Distribution
The Lindley (θ) distribution has the following p.d.f. (Lindley, 1958): The random variable Λ is distributed as Lindley (θ), the marginal distribution of the random variable Y is distributed as PL (θ,ξ) which is: (2)

Bivariate Poisson-Lindley (BPL) Distribution
By setting ξ = 1 in (2), the p.m.f. of PL (θ) distribution which is obtained in Sankaran (1970) is: where, g 1 (y 1 ) and g 2 (y 2 ) are bounded functions in y 1 and y 1 respectively. The value of {.} in (4) is non-negative when ( ) In a similar manner, the joint p.m.f. of BPL (θ 1 , θ 2 , α) distribution is defined as: Where: (3). When α = 0, random variables Y 1 and Y 2 are independent, each is distributed as a marginal PL. Therefore, α is the parameter of independence.
The first five moments of BPL (θ 1 , θ 2 , α) distribution are: where, Thus, Using the variance and covariance in (7), the correlation coefficient is: From (8), Y 1 and Y 2 are independent when α = 0 and have positive and negative correlations when α>0 and α<0 respectively.

Parameter Estimation
The moment estimates of BPL (θ 1 , θ 2 , α) distribution can be obtained by equating the mean and covariance in (7)  Following Sankaran (1970), the unique moment estimate of θ t is for α can then be computed using The log likelihood estimates of BPL (θ 1 , θ 2 , α) can be obtained by maximizing the log likelihood in (9). The Fisher Information matrix can be obtained using the negative expectation of the second derivatives of log likelihood.

Several Tests
As mentioned previously, when α = 0, random variables Y 1 and Y 1 are independent, each is distributed as a marginal PL. For testing independence, we can test H 0 : α = 0 against H 1 : α ≠ 0 and the test can be performed using Likelihood Ratio Test (LRT), , where α is the estimate of independence parameter and ˆ( ) Var α is its estimated variance. The Wald statistic is approximately distributed as a chi-square with one degree of freedom. The variance of parameters for BPL distribution can be estimated using the diagonal elements of the inverse of negative Hessian matrix. The elements of Hessian matrix are the second derivatives of log likelihood.
As another alternative, we can also use a score statistic, which is further discussed in Cox and Hinkley (1979). For the score test, we need the score function, U (θ 1 , θ 2 , α = 0) and the expected information matrix, I (θ 1 , θ 2 , α = 0), which can be obtained from the log likelihood.
The score statistic for testing H 0 : α = 0 against H 1 : α ≠ 0 is: Where: And: The entries for When several models are available, one can compare the models' performance based on several likelihood measures. A regularly used measure is Akaike Information Criteria (AIC) which penalizes a model with larger number of parameters and is defined as where ln L denotes the fitted log likelihood and p the number of parameters.
Application Table 1 provides the flight aborts count data from 109 aircrafts, where random variables Y 1 and Y 2 respectively represent the number of flight aborts in the first and second consecutive six months of a one-year period (Mitchell and Paulson, 1981). Most observed frequencies provide (y 1 , 0) and (0, y 2 ) data, indicating negative correlation between y 1 and y 2 . Therefore, we fit BP (Lakshminarayana et al., 1999), BNB (Famoye, 2010) and BPL (Gomez-Deniz et al., 2012) distributions to the data since these distributions can be fitted to bivariate data with positive, zero or negative correlation.
The joint p.m.f. of BP (θ 1 , θ 2 , α) distribution is (Lakshminarayana et al., 1999): The joint p.m.f. of BNB 1 2 1 2 ( , , , , ) a a θ θ α distribution is (Famoye, 2010):   where, is the dispersion parameter. The mean, variance and covariance are Therefore, the correlation coefficient for BP, BNB and BPL distributions can be positive, zero or negative, depending on the value of multiplicative factor parameter, α. For comparison purpose, Table 1 also provides the fitted values from BP, BNB and BPL distributions. Table 2 provides the estimated parameters, log likelihood and AIC for BP, BNB and BPL distributions. It can be seen that all distributions provide negative value for α, indicating negative correlation. Even though both BNB and BPL distributions produce similar log likelihood, the number of parameters for BPL distribution is less and thus, producing smaller AIC. Based on AIC, BPL distribution provides the best fit for the data.

Conclusion
In this study, BPL distribution has been fitted to a sample of bivariate count data. Based on the results, BPL distribution provides better fit than BP and BNB distributions, indicating that the distribution can be used as an alternative for fitting dependent and over-dispersed count data, with either positive or negative correlation.