INTRODUCTION
The stationary Poisson distribution is a standard model for fitting count data
when the number of occurrences of a phenomenon occurred at a constant rate with
respect to time and an occurrence of the phenomenon does not influence the chance
of any future occurrences. Equality of mean and variance is characteristic of
the Poisson distribution but in a vast number of practical applications, the
count data are either overdispersed or underdispersed (Rainer,
2000).
The Negative Binomial (NB) distribution is another distribution for count data. The NB distribution is often employed in case where a distribution is overdispersed, i.e., its variance is greater than the mean which relaxes the equality of mean and variance property of the Poisson distribution. If X denotes a random variable distributed under a NB distribution with parameter r and p, then its probability mass function (pmf) is given by:
It is well known that:
and
The factorial moment of X is:
where, Γ(.) is the gamma function defined by:
The paper is introduced a new distribution and more flexible alternative to the Poisson distribution when count data are overdispersed in the form of a Negative BinomialBeta Exponential (NBBE) distribution which is a mixed NB distribution obtained by mixing the distribution of NB(r,p) where, p = exp(λ) with distribution of beta exponential (a, b, c). The Beta Exponential (BE) distribution has probability density function (pdf) which has the form:
where, B(.) refers to the beta function defined by:
The Beta Exponential (BE) distribution was introduced by Nadarajah
and Kotz (2006) and it was shown there that its moment generating function
is given by:
In this respect, the aim of this work is to describe our proposed NBBE distribution and show that it includes many well known distributions which are the factorial moments, the first four moments, variance, skewness, kurtosis, parameter estimation of this distribution. The parameters of NBBE distribution are estimated by maximum likelihood method and the usefulness of the NBBE distribution is illustrated by real data set.
THE NEGATIVE BINOMIALBETA EXPONENTIAL (NBBE) DISTRIBUTION
We propose a new mixed NB distribution which is a NBBE distribution obtained by mixing the NB distribution with a BE distribution. We first provide a general definition of this distribution which will subsequently reveal its probability mass function.
Definition 1: Let X be a random variable of a NBBE(r, a, b, c)distribution where X has a NB distribution with parameter r>0 and p = exp(λ) where λ is distributed as BE with positive parameters a, b and c, i.e., Xλ∼NB(r, p = exp(λ)) and λ∼BE(a, b, c).
Theorem 1: Let X∼ NBBE(r, a, b, c). The probability mass function of X is given by:
Proof: If Xλ∼NB(r, p = e^{λ}) in Eq. 1 and λ∼BE(a, b, c) in Eq. 3, then the pmf of X can be obtained by:
where, f (x  λ) is defined by:
By substituting Eq. 7 into Eq. 6, we obtain:
Substituting the moment generating function of BE distribution in Eq. 4 into Eq. 8, the pmf of NBBE (r, a, b, c) is finally given as:
Many well known distributions are subsumed by the NBBE distribution. We display some of these in the next three corollaries and their graphs in Fig. 1.
Corollary 1: If c = 1 then the NBBE distribution reduces to the generalized Waring distribution with pmf given by:
where, r_{(s)} is defined by:
Proof: If Xλ∼NB(r, p = e^{λ}) and λ∼BE(a, b, c = 1), then the pmf of X is:
From Gardshteyn and Ryzhik (2007), the sum of the binomial
terms in Eq. 10 is of the form:
Therefore, h(x) can be written as:

Fig. 1(af): 
The probability mass function of a NBBE random variable (X)
of some values of parameters: (a) r = 1, a = 5, b = 1, c = 1, (b) r = 3,
a = 4, b = 4, c = 5, (c) r = 3, a = 5, b = 2, c = 2, (d) r = 5, a = 5, b
= 10, c = 0.8, (e) r = 20, a = 15, b = 5, c = 5 and (f) r = 30, a = 20,
b = 20, c = 5 
Corollary 2: If a = 1, b = mr and c = 1 then the NBBE distribution reduces to the Waring distribution with pmf given by:
Proof: Substituting a = 1 and b = mr into Eq. 9, then the pmf of X becomes :
Corollary 3: If r = 1, a = 1 and c = 1 then the NBBE distribution reduces to the Yule distribution with pmf given by:
Proof: Substituting r = 1 and a = 1 into Eq. 9, then the pmf of X becomes:
From corollary 13, we find therefore that the generalized Waring distribution
displayed in Eq. 9 (Irwin, 1968; RodriguezAvi
et al., 2009; Wang, 2011), Waring distribution
displayed in Eq. 12 (Irwin, 1975)
and Yule distribution displayed in Eq. 13 (Xekalaki,
1983) are all special cases of the NBBE distribution.
PROPERTIES OF THE NBBE DISTRIBUTION
The first result of this section gives the factorial moment of the NBBE distribution.
Its subsequent corollaries complement the previous corollaries insofar as they
give the corresponding factorial moments of the distributions discussed there.
We hardly need to emphasize the necessity and importance of factorial moment
in any statistical analysis especially in applied work. Some of the most important
features and characteristics of a distribution can be studied through factorial
moments (e.g., mean, variance, skewness and kurtosis).
Theorem 2: If X∼ NBBE(r, a, b, c), then the factorial moment of order k of X is given by:
Proof: If Xλ∼NB(r, p = e^{λ}) and λ∼BE(a,
b, c), then the factorial moment of order k of X can be obtained by:
Using the factorial moment of order k of a negative binomial distribution in Eq. 2, μ_{[k]}(x) becomes:
A binomial expansion of (e^{λ}1)^{k}, then shows that μ_{[k]}(x) can be written as:
From the moment generating function of BE distribution in Eq. 4 with t = kj, we have finally that μ_{[k]}(x) can be written as:
Corollary 4: If c = 1 then the factorial moments of negative binomialbeta exponential reduces to:
which is the same as the factorial moment of order k of generalized Waring distribution.
Proof: Substituting c = 1 into Eq. 14, we get:
Using the expansion Eq. 11 and 16 reduces
to:
Corollary 5: If a = 1, b = mr and c = 1 then the factorial moments of negative binomialbeta exponential reduces to:
which is the same as the factorial moment of order k of Waring distribution.
Proof: Substituting a = 1 and b = mr into Eq. 15, we get:
Corollary 6: If r = 1, a = 1 and c = 1 then the factorial moments of negative binomialbeta exponential reduces to:
which is the same as the factorial moment of order k of the Yule distribution.
Proof: Substituting r = 1 and a = 1 into Eq. 15, we get:
From the factorial moments of NBBE distribution, it is straightforward to
deduce the first four moments given in Eq. 1922,
variance in Eq. 23, skewness in Eq. 24
and kurtosis in Eq. 25:
when b>2/c:
where, Ψ is defined by:
when b>3/c, and:
where, Ω is defined by:
when b>4/c.
PARAMETERS ESTIMATION
The estimation of parameters for NBBE distribution via the Maximum Likelihood Estimation (MLE) method procedure will be discussed.
The likelihood function of the NBBE(r, a, b, c) is given by:
with corresponding loglikelihood function:
The first order conditions for finding the optimal values of the parameters obtained by differentiating Eq. 27 with respect to r, a, b and c give rise to the following differential equations:
and
Table 1: 
Observed and expected frequencies for the accident data 

Equating Eq. 2831 to zero, the MLE solutions
of
and
can be obtained by solving the resulting equations simultaneously using a numerical
procedure such as the NewtonRaphson method.
AN ILLUSTRATIVE EXAMPLE
We used a real data set which number of injured from the accident on major road in Bangkok of Thailand in 2007. The data was collected by Department of Highways, Ministry of Transport, Thailand. We use a real data are fitted by the Poisson distribution, NB distribution and NBBE distribution in Table 1. It show the observed and expected frequencies, grouped in classes of expected frequency greater than five for the chisquare goodness of fit test. The maximum likelihood method provides very poor fit for the Poisson distribution and the NB and acceptable fits for the NBBE.
CONCLUSIONS
We introduced the NBBE distribution which is obtained by mixing the NB distribution
with a BE distribution. We showed that the generalized Waring distribution,
Waring distribution and Yule distribution are all special cases of this distribution.
We have derived the key moments of the NBBE distribution which includes the
factorial moments, mean, variance, skewness and kurtosis. Parameters estimation
are also implemented using maximum likelihood method and the usefulness of the
NBBE distribution is illustrated by real data set. We hope that NBBE distribution
may attract wider applications in analyzing count data.
ACKNOWLEDGMENT
We are grateful to the Commission on Higher Education, Ministry of Education, Thailand, for funding support under the Strategic Scholarships Fellowships Frontier Research Network.