This document presents a new four-parameter negative binomial-Lindley distribution for modeling count data that exhibits overdispersion and has a large number of zeros. The distribution is a generalized version of the negative binomial-Lindley distribution. Statistical properties of the proposed distribution such as the kth factorial moment and dispersion index are derived. Maximum likelihood estimation is used to estimate the parameters. The distribution is fitted to three overdispersed datasets with many zeros, and is shown to outperform Poisson and negative binomial distributions based on model fit statistics.

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A four-parameter negative binomial-Lindley
distribution for modeling over and underdispersed
count data with excess zeros
Razik Ridzuan Mohd Tajuddin, Noriszura Ismail, Kamarulzaman Ibrahim &
Shaiful Anuar Abu Bakar
To cite this article: Razik Ridzuan Mohd Tajuddin, Noriszura Ismail, Kamarulzaman Ibrahim
& Shaiful Anuar Abu Bakar (2020): A four-parameter negative binomial-Lindley distribution for
modeling over and underdispersed count data with excess zeros, Communications in Statistics -
Theory and Methods, DOI: 10.1080/03610926.2020.1749854
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2. A four-parameter negative binomial-Lindley distribution for
modeling over and underdispersed count data with
excess zeros
Razik Ridzuan Mohd Tajuddina
, Noriszura Ismaila
, Kamarulzaman Ibrahima
, and
Shaiful Anuar Abu Bakarb
a
Department of Mathematical Sciences, Universiti Kebangsaan Malaysia, Malaysia; b
Institute of
Mathematical Sciences, University of Malaya, Malaysia
ABSTRACT
Count data often exhibits the property of dispersion and have large
number of zeros. In order to take these properties into account, a new
generalized negative binomial-Lindley distribution with four parame-
ters is proposed, of which the two-parameter and three-parameter
negative binomial-Lindley distributions are special cases. Several statis-
tical properties of the proposed distribution are presented. The disper-
sion index for the proposed distribution is derived and based on the
index, it is clear that the proposed distribution can adequately fit the
data with properties of overdispersion or underdispersion depending
on the choice of the parameters. The proposed distribution is fitted to
three overdispersed datasets with large proportion of zeros. The best
fitted model is selected based on the values of AIC, mean absolute
error and root mean squared error. From the model fittings, it can be
concluded that the proposed distribution outperforms Poisson and
negative binomial distributions in fitting the count data with overdis-
persion and large number of zeros.
ARTICLE HISTORY
Received 19 May 2019
Accepted 26 March 2020
KEYWORDS
Discrete distribution;
dispersed count data; large
number of zeros; mixed
negative binomial
1. Introduction
A mixture of exponential and gamma distribution, which is known as Lindley distribu-
tion has been introduced by Lindley (1958). Several researchers have explored new
count distribution by considering Lindley distribution as the mixing distribution with
the Poisson and negative binomial distributions. For example, a discrete Poisson-
Lindley distribution has been proposed by Sankaran (1970) as an alternative to Poisson
and negative binomial distributions in fitting count data. The estimation of the param-
eter for the discrete Poisson-Lindley distribution using method of moments and max-
imum likelihood estimation technique have been studied extensively by Ghitany and Al-
Mutairi (2009). Recently, Bhati, Sastry and Qadri (2015) have proposed a new general-
ized Poisson-Lindley distribution with applications in the area of insurance. In their
applications, the authors proceeded to estimate the premium for an insurance policy
under Bonus-Malus system.
CONTACT Razik Ridzuan Mohd Tajuddin razikridzuan@siswa.ukm.edu.my Department of Mathematical Sciences,
Universiti Kebangsaan Malaysia, Malaysia.
ß 2020 Taylor & Francis Group, LLC
COMMUNICATIONS IN STATISTICS—THEORY AND METHODS
https://doi.org/10.1080/03610926.2020.1749854

3. Zamani and Ismail (2010) have proposed a two-parameter negative binomial-Lindley
distribution by mixing Lindley distribution with negative binomial distribution, follow-
ing the work of Sankaran (1970). They found that the two-parameter negative bino-
mial-Lindley distribution is particularly suitable in explaining count data with excess
zeros, based on the application to accident and insurance claims data. The work of
Zamani and Ismail (2010) has been further explored by Lord and Geedipally (2011)
based on the simulation study and application on accident data. In addition, as an
extension to the two-parameter negative binomial-Lindley distribution which has been
proposed by Zamani and Ismail (2010), Denthet, Thongteeraparp and Bodhisuwan
(2016) have proposed a three-parameter negative binomial-Lindley distribution by com-
bining two-parameter Lindley distribution proposed by Shanker, Sharma and Shanker
(2013) with negative binomial distribution. Denthet, Thongteeraparp and Bodhisuwan
(2016) found that the new three-parameter negative binomial-Lindley distribution can
fit overdispersed data with large number of zeros, as illustrated from the data on the
number of hospital stays.
When modeling count data, two properties of the data that need to be taken into
account are dispersion and excess zeros. Other than the two and three-parameter nega-
tive binomial-Lindley distributions, many researchers have proposed other mixed nega-
tive binomial distributions which can adequately fit count data with large number of
zeros, such as negative binomial-Erlang (Kongrod, Bodhisuwan and Payakkapong 2014),
negative binomial-generalized exponential (Aryuyuen and Bodhisuwan 2013) and beta-
negative binomial (Wang 2011). On the other hand, to handle overdispersion in count
data, G
omez-D
eniz, Sarabia and Calder
ın-Ojeda (2008) have proposed negative bino-
mial-inverse Gaussian distribution. Generally, the applicability of these mixed negative
binomial is illustrated in fitting insurance and accident data.
In this study, a new distribution, namely a four-parameter negative binomial-Lindley
distribution is proposed and described. The proposed distribution is a generalized ver-
sion of the negative binomial-Lindley distribution, which is an extension to the work
done by Zamani and Ismail (2010) as well as Denthet, Thongteeraparp and Bodhisuwan
(2016). The four-parameter negative binomial-Lindley distribution is the combination of
the negative binomial distribution and the three-parameter Lindley distribution. This
paper is arranged in the following order. The statistical properties for this new distribu-
tion such as the kth
factorial moment and dispersion index are obtained and explained
in Section 2. Section 3 describes the parameter estimation of four-parameter negative
binomial-Lindley distribution using maximum likelihood estimation method. Section 4
consists of three applications of the four-parameter negative binomial-Lindley distribu-
tion and illustration of model fitting based on Poisson, negative binomial and the pro-
posed distributions. Section 5 concludes the study.
2. Probability mass function and some properties
A mixed Poisson-gamma distribution, which is known as negative binomial distribution,
is often used for modeling overdispersed count data, as an alternative to Poisson distri-
bution. The probability mass function (pmf) for a random variable X which follows a
negative binomial distribution is given as
2 R. R. M. TAJUDDIN ET AL.

4. Pr X ¼ xjr, p
¼
x þ r 1
x
pr
1 p
ð Þx
; x ¼ 0, 1, 2, 3, :::
where r 0 and 0 p 1: By taking p ¼ e k
where k is a random variable which fol-
lows a certain distribution, where k 0, the pmf of X can be written as
Pr X ¼ xjr, k
ð Þ ¼
x þ r 1
x
e kr
1 e k
ð Þx
; x ¼ 0, 1, 2, 3, ::: ð1Þ
Definition 1: A random variable X follows a four-parameter negative binomial-Lindley
distribution if it obeys the following stochastic representation:
Xjr, k NB r, p ¼ e k
and
Figure 1. The graphs of probability function of NBL3 distribution when (i) r ¼ 1, a ¼ 1, b ¼ 1 and h
varies, (ii) r ¼ 1, h ¼ 1, b ¼ 1 and a varies, (iii) r ¼ 1, a ¼ 1, h ¼ 1 and b varies, (iv) h ¼ 1, a ¼
1, b ¼ 1 and r varies.
COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 3

5. k f k
ð ja, b, hÞ ¼
h2
ha þ b
a þ bk
ð Þe hx
where
k 0, a 0, b 0 and h 0: The f k
ð ja, b, hÞ distribution refers to the three-parameter
Lindley distribution, which have been proposed by Shanker et al. (2017) and will be
denoted as L3 a, b, h
ð Þ: The four-parameter negative binomial-Lindley distribution will
be represented as NBL3: The shape of the probability value for a random variable X
which follows NBL3 with different parameters values are plotted and given in Figure 1.
The graphs are plotted to investigate the effect of certain parameters to the probability
value given that the other parameters are set to one.
Based on Figure 1, the ability of the distribution in fitting data with large number of
zeros can be identified. Specifically, when parameter h is large and the other parameters
are set to one, the probability of an event not happening is significantly high. The same
observation can be made when the dispersion r is small. As a increases and the other
parameters are set to one, the probability value reduces to zero quickly. The same con-
clusion can be made when b decreases and the other parameters are set to one. Overall,
a and b are the weight parameters, h is the rate parameter whereas r is the dispersion
parameter for NBL3 distribution. In conclusion, the NBL3 distribution is an alternative
distribution to adequately fit count data with excess zeros.
Theorem 1: Let X be a random variable which follows a four-parameter negative bino-
mial-Lindley distribution with parameters r, a, b and h: Then, the pmf of X is given by
Pr X ¼ xjr, a, b, h
ð Þ ¼
x þ r 1
x
!
h2
ah þ b
X
x
j¼0
x
j
!
1
ð Þj a h þ r þ j
ð Þ þ b
h þ r þ j
ð Þ2
#
; x ¼ 0, 1, ::: 2
ð Þ
where a 0, b 0 and h 0:
Proof: If Xjr, k NBðr, p ¼ e k
Þ and k L3 a, b, h
ð Þ, then the marginal distribution for
X can be obtained using
PrðX ¼ xjr, a, b, hÞ ¼
ð1
0
Pr X ¼ xjr, k
ð Þf kja, b, h
ð Þdk
Know that
1 e k
ð Þx
¼
X
x
j¼0
x
j
1
ð Þj
e kj
Therefore, equation (1) can be written as
Pr X ¼ xjr, k
ð Þ ¼
x þ r 1
x
X
x
j¼0
x
j
1
ð Þj
e k rþj
ð Þ
By using marginal distribution formula above, the pmf of the NBL3 distribution can
be obtained as
4 R. R. M. TAJUDDIN ET AL.

6. PrðX ¼ xjr, a, b, hÞ ¼
x þ r 1
x
!
X
x
j¼0
x
j
!
1
ð Þj
ð1
0
e k rþj
ð Þ
f kja, b, h
ð Þdk
¼
x þ r 1
x
X
x
j¼0
x
j
1
ð Þj
Mk r þ j
ð Þ
½ Š
where Mk t
ð Þ is the moment generating function (mgf) of L3 a, b, h
ð Þ, given in Lemma 1.
Lemma 1: Let k be a random variable which follows a three-parameter Lindley distribu-
tion with parameters a, b and h: Then, the mgf of k is given by
Mk t
ð Þ ¼ St=S0 ð3Þ
where
St ¼
a h t
ð Þ þ b
h t
ð Þ2
Proof:
Mk t
ð Þ ¼ E etk
ð Þ ¼
ð1
0
etk h2
ha þ b
a þ bk
ð Þe hk
dk
¼
h2
ha þ b
a
ð1
0
e k h t
ð Þ
dk þ b
ð1
0
ke k h t
ð Þ
dk
¼
h2
ha þ b
a
1
h t
þ b
1
h t
ð Þ2
#
¼
h2
ha þ b
a h t
ð Þ þ b
h t
ð Þ2
#
¼ St=S0
Thus, the pmf of NBL3 distribution can be written as in equation (2) by substituting back
into the equation prior to equation (3). Several special cases based on NBL3 distribution are
summarized in Table 1. Overall, it can be concluded that NBL3 distribution is a versatile dis-
tribution as it nests several distributions when certain values of parameters are fixed.
Theorem 2: Let X be a random variable which follows a four-parameter negative bino-
mial-Lindley distribution with parameters r, a, b and h: Then, the kth
factorial moment
of X is given by
Table 1. Special cases of NBL3 distribution under certain conditions.
Condition Distribution
a ¼ 1 A three-parameter negative binomial-Lindley distribution,
NBL2 r, b, h
ð Þ [4].
a ¼ b ¼ g; g 2 Rþ
A two-parameter negative binomial-Lindley distribution,
NBL r, h
ð Þ [2].
a ! 0þ
and r, b, h remain constant OR
b ! 1 and r, a, h remain constant
A special case of negative binomial-Erlang distribution,
NB ELðr, k ¼ 2, cÞ [6].
b ! 0þ
and r, a, h remain constant OR
a ! 1 and r, b, h remain constant
A special case of negative binomial-Erlang distribution,
NB ELðr, k ¼ 1, cÞ [6] and a special case of negative
binomial-generalized exponential distribution,
NB GEðr, a ¼ 1, bÞ [7].
r ¼ 1 A three-parameter geometric-Lindley distribution.
COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 5

7. l0
ðkÞ ¼
C r þ k
ð Þ
C r
ð Þ
h2
ah þ b
!
X
k
j¼0
k
j
1
ð Þj a h k þ j
ð Þ þ b
h k þ j
ð Þ2
#
Proof: For a mixture of negative binomial distribution, the kth
factorial moment can be
obtained by using equation (4) that has been employed by many researchers (Aryuyen
and Bodhisuwan 2013; Denthet, Thongteeraparp and Bodhisuwan 2016; G
omez-D
eniz,
Sarabia and Calder
ın-Ojeda 2008; Kongrod, Bodhisuwan and Payakkapong 2014;
Zamani and Ismail 2010).
l0
ðkÞ ¼
C r þ k
ð Þ
C r
ð Þ
Ek ek
1
ð Þk
(4)
By employing binomial expansion, equation (4) can be written as
l0
k
ð Þ ¼
C r þ k
ð Þ
C r
ð Þ
X
k
j¼0
k
j
1
ð Þj
Ek ek k j
ð Þ
½ Š ¼
C r þ k
ð Þ
C r
ð Þ
X
k
j¼0
k
j
1
ð Þj
Mk k j
ð Þ (5)
By substituting equation (3) into equation (5), the kth
factorial moment can be acquired
as in Theorem 2. And thus, the kth
moment about origin can also be obtained. The first
two factorial moments are
l ¼ l0
1
ð Þ ¼
r
S0
S1 S0
ð Þ
and
l0
2
ð Þ ¼
r r þ 1
ð Þ
S0
S2 2S1 þ S0
ð Þ
respectively. By using the relationship l2 ¼ l0
2
ð Þ þ l, the second moment about origin
can be written as
l2 ¼
r
S0
r þ 1
ð Þ S2 S1
ð Þ r S1 S0
ð Þ
Based on the two moments about the origin, the variance and dispersion can be
obtained. In order to know whether NBL3 can fit overdispersed, underdispersed data or
both, dispersion index needs to be found. Since both first and second moment are not
in a simple form, the dispersion index, d formula is found using the relation d ¼
r2
=l ¼ l2=l l, given as
d ¼ r þ 1
ð Þ
S2 S1
S1 S0
r
S1
S0
6
ð Þ
where
S2 ¼
a h 2
ð Þ þ b
h 2
ð Þ2 , S1 ¼
a h 1
ð Þ þ b
h 1
ð Þ2 , S0 ¼
ah þ b
h2 , h 6¼ 0, 1, 2:
Equation (6) cannot be directly used to determine whether the NBL3 distribution can
adequately fit overdispersed or underdispersed data. Therefore, dispersion values are
6 R. R. M. TAJUDDIN ET AL.

8. plotted as a function of parameters in Figure 2, to show the ability of the proposed dis-
tribution in describing data with the properties of either overdispersion or underdisper-
sion, given that certain parameters are set to one. Based on plots (i), (ii) and (iii) in
Figure 2, it is clear that the dispersion index of the proposed distribution can be either
less than one or greater than one depending on the choice of the parameters. The plot
(iv) in Figure 2 shows that as the value of h increases, the dispersion index approaches
to one.
3. Maximum likelihood estimator
The MLEs are the estimators that maximize the log-likelihood value of NBL3 distribu-
tion. The log-likelihood function, l in NBL3 distribution is given as
Figure 2. The graphs of dispersion values for NBL3 distribution when (i) b ¼ 1 and r, a and h vary,
(ii) a ¼ 1 and r, b and h vary, (iii) a ¼ 1, b ¼ 1, h and r vary, (iv) a ¼ 1, b ¼ 1, h and r vary.
COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 7

9. l ¼ ln L r,a,b,h
ð Þ ¼
X
n
i¼1
ln Pr Xi ¼ xijr,a,b,h
ð Þ
½ Š ¼
X
1
x¼0
nxln Pr X ¼ xjr,a,b,h
ð Þ
½ Š
¼
X
1
x¼0
nx ln
x þ r 1
x
!
þ 2lnh ln ah þ b
ð Þ þ ln
X
x
j¼0
x
j
!
1
ð Þj a h þ r þ j
ð Þ þ b
h þ r þ j
ð Þ2
2
4
3
5
8
:
9
=
;
where nx refers to the frequency for x-valued data. The first partial derivative with
respect to all four parameters are given as
@l
@h
¼ n
2
h
a
ahþ b
þ
X
1
x¼0
nx
Px
j¼0
x
j
1
ð Þjþ1 a hþrþj
ð Þþ2b
hþrþj
ð Þ3
Px
j¼0
x
j
1
ð Þj a hþrþj
ð Þþb
hþrþj
ð Þ2
2
6
6
6
4
3
7
7
7
5
(7)
@l
@a
¼
nh
ah þ b
þ
X
1
x¼0
nx
Px
j¼0
x
j
1
ð Þjþ1 1
hþrþj
Px
j¼0
x
j
1
ð Þj a hþrþj
ð Þþb
hþrþj
ð Þ2
2
6
6
6
4
3
7
7
7
5
(8)
@l
@b
¼
n
ah þ b
þ
X
1
x¼0
nx
Px
j¼0
x
j
1
ð Þjþ1 1
hþrþj
ð Þ2
Px
j¼0
x
j
1
ð Þj a hþrþj
ð Þþb
hþrþj
ð Þ2
2
6
6
6
4
3
7
7
7
5
(9)
@l
@r
¼
@
@r
X
1
x¼0
nx ln
x þ r 1
x
( )
þ
@
@r
X
1
x¼0
nxln
X
x
j¼0
x
j
1
ð Þj a h þ r þ j
ð Þ þ b
h þ r þ j
ð Þ2
#
( )
(10)
Let the expression in the first term of the partial derivative with respect to r, @l=@r
in (10) be represented as
A r
ð Þ ¼
X
1
x¼0
nx ln
x þ r 1
x
The derivative of the expression, AðrÞ can be written as (Klugman, Panjer Willmot
2008)
@
@r
A r
ð Þ
½ Š ¼
@
@r
X
1
x¼0
nx ln
x þ r 1
x
( )
¼
X
1
x¼0
nx
@
@r
ln
x þ r 1
ð Þ x þ r 2
ð Þ:::r
x!
Some simplification of the equation above gives
@
@r
A r
ð Þ
½ Š ¼
X
1
x¼0
nx
@
@r
ln
Y
x 1
m¼0
r þ m
ð Þ ¼
X
1
x¼0
nx
@
@r
X
x 1
m¼0
ln r þ m
ð Þ ¼
X
1
x¼0
nx
X
x 1
m¼0
1
r þ m
11
ð Þ
8 R. R. M. TAJUDDIN ET AL.

10. Equation (10) can be re-written by substituting equation (11) as
@l
@r
¼
X
1
x¼0
nx
X
x 1
m¼0
1
r þ m
þ
X
1
x¼0
nx
Px
j¼0
x
j
1
ð Þjþ1 a hþrþj
ð Þ2
þ2b hþrþj
ð Þ
hþrþj
ð Þ4
Px
j¼0
x
j
1
ð Þj a hþrþj
ð Þþb
hþrþj
ð Þ2
2
6
6
6
4
3
7
7
7
5
(12)
All four equations (7) (8) (9) and (12) can be solved using iterative method until the
estimates converge. The best models are chosen based on the Akaike’s Information
Criterion (AIC), mean absolute error (MAE) and root mean squared error (RMSE). The
formula for AIC (Akaike 1974), MAE and RMSE are given by
AIC ¼ 2l þ 2k
MAE ¼
1
n
X
n
i¼1
yi ^
yi
and
RMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
n
X
n
i¼1
yi ^
yi
2
s
respectively, where k refers to number of estimated parameters and ^
yi is the fitted values
of the ith
data. The model that gives the smallest AIC, MAE and RMSE values in the
model fitting will be selected as the best model.
4. Applications
The NBL3 distribution is fitted to several datasets and the results of the fittings are com-
pared with those found from the fittings of Poisson, denoted as Pois and negative bino-
mial, denoted as NB distributions.
Table 2. Distribution of the crash count (2003-2008) obtained by fitting Poisson, negative binomial
and four-parameter negative binomial-Lindley distributions.
x n
Distributions
Pois NB NBL3
0 29,087 28,471.92 29,101.87 29,099.79
1 2,952 3,917.74 2,858.87 2904.85
2 464 269.54 549.34 499.25
3 108 12.36 122.76 116.44
4 40 0.43 29.35 33.32
5 9 0.01 7.29 11.07
6þ 12 0.00 2.52 7.28
Total 32,672 32,672 32,672 32,672
Parameter estimates ^
k ¼ 0.1376 ^
p ¼ 0.2861 ^
h ¼ 9.6762
^
r ¼ 0.3434 ^
r ¼ 1.1444
^
a ¼ 2.9289
^
b ¼ 1.1129
l 14,208.06 13,549.61 13,528.77
AIC 28,418.12 27,103.22 27,065.54
MAE 175.59 21.18 10.65
RMSE 351.57 38.77 18.48
COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 9

11. Example 1 The first dataset is the crash data that has been considered by Lord and
Geedipally (2011). The crash data refers to the single-vehicle roadway departure crashes
on rural two-lane horizontal curves in Texas between 2003 and 2008. The dispersion of
the data, calculated as the ratio of variance to mean is 1.49 and an 89% of the data takes
on value of zero. The results of model fittings for the number of crashes are summar-
ized in Table 2. Based on Table 2, the model fitting of NBL3 distribution gives the
smallest AIC, MAE and RMSE values, and thus is selected as the best model with fitted
function given as
Pr X ¼ xj^
r, ^
a, ^
b, ^
h ¼
x þ ^
r 1
x
^
h
2
^
a^
h þ ^
b
X
x
j¼0
x
j
1
ð Þj
^
a ^
h þ ^
r þ j þ ^
b
^
h þ ^
r þ j
2
2
6
4
3
7
5
where ^
r ¼ 1:1444, ^
a ¼ 2:9289, ^
b ¼ 1:1129, ^
h ¼ 9:6762:
Example 2 The second dataset refers to the length of stays in hospital for American res-
idents age 66 years and above. This data has been considered by Denthet,
Thongteeraparp and Bodhisuwan (2016). The dispersion of the data, calculated as the
ratio of variance to mean is 1.88 and an 80% of the data takes on value of zero. The
results of model fittings for the length of stays in hospital for American residents age 66
years and above are summarized in Table 3.
Based on Table 3, the fitting of NB distribution gives the smallest AIC value.
However, the AIC values from the fittings of NB and NBL3 distributions do not differ
that much. In addition, the MAE and RMSE values from the fitting of NBL3 distribu-
tion is smaller than that from NB distribution. Therefore, NBL3 distribution is the best
model in describing the length of stays in hospital for American residents age 66 years
and above, with fitted function given as
Table 3. Distribution of the length of stays in hospital for American residents age 66 years and
above obtained by fitting Poisson, negative binomial and four-parameter negative binomial-Lindley
distributions.
x n
Distributions
Pois NB NBL3
0 3,541 3,277.13 3,544.31 3,538.77
1 599 970.03 583.54 609.24
2 176 143.56 177.52 162.33
3 48 14.17 62.26 54.63
4 20 1.05 23.29 21.47
5 12 0.06 9.03 9.45
6þ 10 0.00 6.05 10.09
Total 4,406 4,406 4,406 4,406
Parameter estimates ^
k ¼ 0.2960 ^
p ¼ 0.4438 ^
h ¼ 6.7802
^
r ¼ 0.3710 ^
r ¼ 0.9249
^
a ¼ 0.4689
^
b ¼ 8.4641
l 3304.51 3009.63 3007.74
AIC 6611.02 6023.26 6023.48
MAE 82.45 5.07 4.37
RMSE 152.76 7.35 6.25
10 R. R. M. TAJUDDIN ET AL.

12. Pr X ¼ xj^
r, ^
a, ^
b, ^
h ¼
x þ ^
r 1
x
^
h
2
^
a^
h þ ^
b
X
x
j¼0
x
j
1
ð Þj
^
a ^
h þ ^
r þ j þ ^
b
^
h þ ^
r þ j
2
2
6
4
3
7
5
where ^
r ¼ 0:9249, ^
a ¼ 0:4689, ^
b ¼ 8:4641, ^
h ¼ 6:7802:
Example 3 The third dataset is the number of claims of the third liability vehicle insur-
ance considered in the study by Wang (2011). The dispersion of the data, calculated as
the ratio of variance to mean is 1.55 and an 77% of the data takes on value of zero.
Table 4 refers to the parameters estimates and fitted values of the crash data with all
three distributions.
Based on Table 4, the NBL3 distribution is the best model in describing the number
of claims of the third liability vehicle insurance as the model gives the smallest AIC,
MAE and RMSE values, and thus is selected as the best model with fitted function given
as
Pr X ¼ xj^
r, ^
a, ^
b, ^
h ¼
x þ ^
r 1
x
^
h
2
^
a^
h þ ^
b
X
x
j¼0
x
j
1
ð Þj
^
a ^
h þ ^
r þ j þ ^
b
^
h þ ^
r þ j
2
2
6
4
3
7
5
where ^
r ¼ 3:6714, ^
a ¼ 5:7606, ^
b ¼ 4:6852, ^
h ¼ 13:2776:
5. Conclusions
The proposed distribution which is named NBL3 distribution, is a generalization for
two and three-parameter negative binomial-Lindley distributions. The proposed distri-
bution is quite general where its special cases resulted in several types of mixed negative
binomial distributions such as negative binomial-Erlang and negative binomial-general-
ized exponential distributions. Some statistical properties for the proposed model has
Table 4. Distribution of the number of claims of the third liability vehicle insurance obtained by fit-
ting Poisson, negative binomial and four-parameter negative binomial-Lindley distributions.
x n
Distributions
Pois NB NBL3
0 27,141 25,528.69 27,165.77 27,131.38
1 5,789 8,107.91 5,664.18 5,789.58
2 1,443 1,287.54 1,563.33 1,483.31
3 457 136.31 466.66 436.83
4 155 10.82 144.55 143.70
5 56 0.69 45.75 51.76
6þ 31 0.04 21.76 35.44
Total 35,072 35,072 35,072 35,072
Parameter estimates ^
k ¼ 0.3176 ^
p ¼ 0.6565 ^
h ¼ 13.2776
^
r ¼ 0.6070 ^
r ¼ 3.6714
^
a ¼ 5.7606
^
b ¼ 4.6852
l 26,712.72 25,422.52 25,418.50
AIC 53,427.44 50,849.04 50,845.00
MAE 463.78 31.62 10.44
RMSE 901.58 55.71 15.40
the best model is written in bold
COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 11

13. been proposed to understand the NBL3 distribution. The properties studied include the
kth
factorial moment and the dispersion index. Based on these statistical properties, one
can easily obtain the measures of skewness, kurtosis and higher order moments for
NBL3: From the dispersion index, it can be concluded that the data generated from the
NBL3 distribution can have the properties of either underdispersion or overdispersion.
The derivation of maximum likelihood estimators of the parameters of NBL3 distribu-
tion is also presented. Even though NBL3 distribution has four parameters, the adequacy
of the model for NBL3 is significantly improved compared to those for Poisson and
negative binomial, suggesting that the NBL3 can be considered in fitting dispersed count
data with large number of zeros.
Acknowledgments
The authors would also like to thank the referees for the constructive comments.
Funding
The authors gratefully acknowledge the financial support received through research grants
(FRGS/1/2019/STG06/UKM/01/5) from Ministry of Education, Malaysia and (GUP-2019-031)
from Universiti Kebangsaan Malaysia.
ORCID
Razik Ridzuan Mohd Tajuddin http://orcid.org/0000-0001-6534-3678
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