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.
Highlights
Mixed Poisson and mixed negative binomial distributions have been considered as alternatives for fitting count data with overdispersion
Examples of mixed Poisson and mixed negative binomial distributions are Negative Binomial (NB) obtained as a mixture of Poisson and gamma, Poisson-Lindley (PL) (Sankaran, 1970; Ghitany et al, 2008), Poisson-Inverse Gaussian (PIG) (Trembley, 1992; Willmot, 1987), Negative BinomialPareto (NBP) (Meng et al, 1999), Negative BinomialInverse Gaussian (NBIG) (Gomez-Deniz et al, 2008), negative binomial-Lindley (NBL) (Zamani and Ismail, 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 Bivariate Negative Binomial (BNB), Bivariate PoissonLognormal (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
Summary
Mixed Poisson and mixed negative binomial distributions have been considered as alternatives for fitting count data with overdispersion. 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 correlation coefficient for BP, BNB and BPL distributions can be positive, zero or negative, depending on the value of multiplicative factor parameter, α. 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 producing smaller AIC. Based on AIC, BPL distribution provides the best fit for the data
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