Journal of Mathematics and Statistics Original Research Paper
Bivariate Poisson-Lindley Distribution with Application 1
Hossein Zamani, 2Pouya Faroughi and 3Noriszura Ismail
1
Department of Statistics, Hormozgan University, Bandarabbas, Iran Department Statistics, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran 3 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia 2
Article history Received: 25-09-2014 Revised: 31-12-2014 Accepted: 27-01-2015 Corresponding Author: Noriszura Ismail School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia Email:
[email protected]
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.
Keywords: Bivariate, Poisson-Lindley, Dependent, Over-Dispersed, Count Data Besides mixture approach, several bivariate discrete Introduction distributions have been defined using the method of Mixed Poisson and mixed negative binomial trivariate reduction (Kocherlakota and Kocherlakota, 1999; distributions have been considered as alternatives for Johnson et al., 1997). The BP distribution from the fitting count data with overdispersion. Examples of mixed trivariate reduction has been used for modeling Poisson and mixed negative binomial distributions are correlated bivariate count data and several applications Negative Binomial (NB) obtained as a mixture of Poisson can be found in (Holgate, 1964; Paul and Ho, 1989). and gamma, Poisson-Lindley (PL) (Sankaran, 1970; Besides BP distribution, the Bivariate Generalized Ghitany et al., 2008), Poisson-Inverse Gaussian (PIG) Poisson (BGP) distribution from the trivariate reduction has (Trembley, 1992; Willmot, 1987), Negative Binomialbeen defined and studied in Famoye and Consul (1995), Pareto (NBP) (Meng et al., 1999), Negative Binomialwhere the distribution can be used for modeling Inverse Gaussian (NBIG) (Gomez-Deniz et al., 2008), correlated and under- or overdispersed bivariate count data. negative binomial-Lindley (NBL) (Zamani and Ismail, 2010; Lord and Geedipally, 2011) and Poisson-Weighted In this study, we apply the BPL distribution which was Exponential (PWE) (Zamani et al., 2014a) distributions. derived from the product of two PL marginals with a Based on literatures, the mixture approaches have been multiplicative factor parameter. This BPL distribution can used to derive new families of bivariate distribution. The be used for bivariate count data with positive, zero or Bivariate Negative Binomial (BNB), Bivariate Poissonnegative correlation coefficient. The rest of this study is Lognormal (BPLN), Bivariate Poisson-Inverse Gaussian organized as follows. Section 2 provides the univariate (BPIG) and bivariate Poisson-Lindley (BPL) distributions version of PL distribution. Several properties of the BPL are several examples of classes of mixed distribution which distribution, such as mean, variance and correlation are extended from univariate case. For further literatures, coefficient, are discussed in section 3. Section 4 discusses BNB distribution was studied in Marshall and Olkin (1990) parameter estimation for the BPL and section 5 provides and applied in Karlis and Ntzoufras (2003), tests for several tests for testing independence. Numerical overdispersion and independence in BNB model were illustration is provided in section 6, where BPL distribution discussed in (Jung et al., 2009; Cheon et al., 2009), BPIG is fitted to the bivariate flight aborts count data. The BPL distribution was derived by Stein et al. (1987), BPL distribution is compared to BP (Lakshminarayana et al., distribution was proposed by Gomez-Deniz et al. (2012) 1999) and BNB (Famoye, 2010) distributions which also and Bivariate Poisson-Weighted Exponential (BPWE) was allow positive, zero or negative correlation. proposed in Zamani et al. (2014b). © 2015 Hossein Zamani, Pouya Faroughi and Noriszura Ismail. This open access article is distributed under a Creative Commons Attribution (CC-BY) 3.0 license.
Hossein Zamani et al. / Journal of Mathematics and Statistics 2015, 11 (1): 1.6 DOI: 10.3844/jmssp.2015.1.6
Poisson marginals with a multiplicative factor parameter, is defined as (Lakshminarayana et al., 1999):
Univariate Poisson-Lindley (PL) Distribution The Lindley (θ) distribution has the following p.d.f. (Lindley, 1958): f ( x) =
θ2 (1 + x )e −θ x , θ +1
Pr(Y1 = y1 ,Y2 = y2 ) = e − λ1 − λ2
x > 0,θ > 0
y1 , y2 = 0,1, 2,...,
θ 2 θ − z +1 θ + 1 (θ − z ) 2
e − λξ (λξ ) y , y!
y = 0,1,2,...,
λ, µ > 0
P(Y1 = y1, Y2 = y2 ) = [1 + α (e
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
Pr(Y = y ) =
λ1 , λ2 > 0
where, g1(y1) and g2(y2) are bounded functions in y1 and y1 respectively. The value of {.} in (4) is non-negative when gt ( yt ) = e − yt and gt = E[ gt (Yt )] = E (e −Yt ), t = 1, 2 . In a similar manner, the joint p.m.f. of BPL (θ1, θ2, α) distribution is defined as:
(1)
Assuming the conditional random variable YΛ follows Poisson distribution with p.m.f: Pr(Y = y | λ ) =
(4)
{1 + α [( g1 ( y1 ) − g1 )( g 2 ( y2 ) − g 2 )]}
And m.g.f: M X (z) =
λ1y1 λ2y2 y1 ! y2 !
− y1
− c1 )(e
− y2
θ12 ( y1 + θ1 + 2) θ 22 ( y2 + θ 2 + 2) (θ1 + 1) y1 + 3 (θ 2 + 1) y2 + 3 − c2 )], y1, y2 = 0,1, 2,...,
(5)
θ1,θ 2 > 0
Where:
y
θ ξ θ + ξ + y +1 , y = 0,1,2,..., θ ,ξ > 0 1 + θ (θ + ξ ) y + 2
(2)
ct = E (e −Yt ) =
θt2 θt + 2 − e −1 , t = 1, 2 1 + θt (θt − e −1 + 1) 2
(6)
Bivariate Poisson-Lindley (BPL) Distribution We obtain E (e −Yt ) in (6) by letting z = −1 in m.g.f. (3). When α = 0, random variables Y1 and Y2 are independent, each is distributed as a marginal PL. Therefore, α is the parameter of independence. The first five moments of BPL (θ1, θ2, α) distribution are:
By setting ξ = 1 in (2), the p.m.f. of PL (θ) distribution which is obtained in Sankaran (1970) is: Pr(Y = y ) =
θ 2 (θ + y + 2) , y = 0,1,2,..., θ > 0 (θ + 1) y + 3
With mean: E (Y t ) = µt =
θ +2 E (Y) = θ (θ + 1)
Variance:
V ar (Y t ) = σ t2 = Var (Y ) =
θ 3 + 4θ 2 + 6θ + 2 θ 2 (θ + 1)2
θt + 2 , θt (θt + 1)
t = 1, 2
θt3 + 4θt2 + 6θt + 2 , θt2 (θt + 1)2
t = 1,2
And:
And m.g.f:
Cov(Y1 , Y2 ) = α (c11 − µ1c1 )(c22 − µ 2c2 )
(7)
M Y ( z ) = M Λ (e z − 1) = E (e ZY ) =
θ 2 θ + 2 − ez 1 + θ (θ − e z + 1) 2
where, ctt = E (Yt e −Yt ), t = 1, 2 differentiating m.g.f. in (3) with respect to z and letting z = −1 , we have
(3)
∂ = E (Ye −Y ) . M Y ( z) ∂z z =−1
In this study, we use the BPL distribution which was derived by Gomez-Deniz et al. (2012), who used the methodology proposed by Lee (1996) and ideas suggested in Sarmanov (1966). The same approach was also used by Lakshminarayana et al. (1999) for deriving BP distribution. The joint p.m.f. of BP (λ1, λ1, α) distribution, which was derived from the product of two
ctt =
Thus,
θt2 θt e −1 − e −2 + 3e −1 , t = 1, 2 . 1 + θt (θt − e −1 + 1)3
Using the variance and covariance in (7), the correlation coefficient is: 2
Hossein Zamani et al. / Journal of Mathematics and Statistics 2015, 11 (1): 1.6 DOI: 10.3844/jmssp.2015.1.6
ρ12 =
σ 12 α (c11 − µ1c1 )(c22 − µ 2c2 ) = σ 1σ 2 σ 1σ 2
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 (θ1, θ2, α = 0) and the expected information matrix, I (θ1, θ2, α = 0), which can be obtained from the log likelihood. The score statistic for testing H0: α = 0 against H1: α ≠ 0 is:
(8)
From (8), Y1 and Y2 are independent when α = 0 and have positive and negative correlations when α>0 and α
