A COM-Poisson type Generalization of the Negative ...

To appear in Communications in Statistics, Theory and Methods

A COM-Poisson type Generalization of the Negative Binomial Distribution S. Chakraborty a, b and S. H. Ongb a

Department of Statistics, Dibrugarh University, Dibrugarh - 786004, India b

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

Abstract This paper introduces a generalization of the negative binomial (NB) distribution in analogy with the COM-Poisson distribution. Many well-known distributions are particular and limiting distributions. The proposed distribution belongs to the modified power series, generalized hypergeometric and exponential families, and also arises as weighted negative binomial and COM-Poisson distributions. Probability and moment recurrence formulae, probabilistic and reliability properties have been derived. With the flexibility to model under, equi- and over dispersion, and its various interesting properties, this NB generalization will be a useful model for count data. An application to empirical modeling is illustrated with a real data set.

Keywords and Phrases COM-Poisson, weighted distribution, index of dispersion, under, equiand over dispersion, reliability, unimodality, log-concavity, increasing failure rate, stochastic ordering, modified power series, generalized hypergeometric, exponential families, empirical modeling. Mathematics Subject Classification (2000) 62E15·62F03·62N05

1. Introduction The Poisson distribution is often used to model count frequency data but the equality of its mean and variance (equi-dispersion) makes it too restrictive for many applications. This is because unequal mean and variance is usually encountered in observed data. Under dispersion (variance less than the mean) occurred relatively less in applications. Examples of under dispersed distributions are the Charlier series distribution and its generalizations (Sugita et al, 2011; Ong et al, 2012). Over dispersion is more prevalent and this has led researchers to consider alternative approaches to generalize or extend the Poisson model. For example, in the method of mixtures (see Gupta and Ong, 2005), the Poisson parameter (mean) is allowed to vary as a random variable resulting in a mixed Poisson distribution which has variance greater than the mean (over dispersion). If this is the gamma random variable, then the negative binomial distribution (Greenwood and Yule, 1920) is obtained. Models for count data that can cater for under, equi- and over dispersion have also attracted the attention of researchers. One such model is the COM-Poisson distribution (Conway and Maxwell, 1962; Shmueli et al, 2005) which has seen a recent revival of interest. Of late, 1

many researchers have studied the COM-Poisson distribution. For example, Nadarajah (2009) derived exact and explicit expressions for its moments and the cumulative distribution function in case of over dispersion while Cordeiro et al (2012), by compounding an exponential distribution with the COM-Poisson distribution, obtained a more general distribution known as the exponential-Conway-Maxwell Poisson distribution. Rodrigues et al (2009) developed a flexible cure rate survival model based on the COM-Poisson distribution, and Cancho et al. (2012) examined Bayesian analysis for this cure rate model. The probability mass function (pmf) of the COM-Poisson distribution is given by pk 1 (1) P( X  k )  , k  0, 1,   ( k !) Z ( p ,  ) 

where Z ( p,  ) 



pj

j 0 ( j! )



for p  0 and   0 . Comparing this with the Poisson pmf, the

COM-Poisson pmf is seen to have an additional parameter  which gives it flexibility for modeling under, equi- and over dispersion. Another distribution with this flexibility is the generalized Poisson distribution (GPD) of Consul (1989). The GPD has pmf

1 k 1  ( a  k z ) for k  0,1,   a (a  kz ) e P ( X  k )   k!  0 for k  m, when z  0

(2)

where a  0 , max (1, a / m)  z 1 and m ( 4) is the largest positive integer satisfying a  mz  0 when a is negative. For under dispersion the parameters have to be restricted in order for the GPD to be a proper distribution (Nelson, 1985). Conway and Maxwell (1962) have derived the COM-Poisson distribution by considering a queuing system with state dependent service rate. Mathematically, the COM-Poisson may be formulated by taking the term k ! in the Poisson pmf to the power of  and normalizing the resulting expression. In this paper we generalize the negative binomial distribution by introducing an additional parameter in the same manner. Many well-known distributions like Poisson, geometric, negative binomial, COM-Poisson are particular cases while the Bernoulli and COM-Poisson distributions are seen as limiting distributions. The new distribution belongs to the modified power series, generalized hypergeometric families and also arises as weighted negative binomial and COM-Poisson distributions. Probability and moment recurrence formulae, index of dispersion, probabilistic and reliability properties like unimodality, log-concavity, increasing failure and stochastic ordering have been derived. Like the COM-Poisson distribution, this COM-Poisson type negative binomial distribution (COM-NB) also has the flexibility to model under, equi and over dispersion. Due to this flexibility and various interesting properties, it is expected that the COM-NB distribution will be a useful addition as a model for count data.

2

The organization of the paper is as follows. Section 2 gives the definition, basic properties, limiting cases and formulations. Reliability and probabilistic properties are presented in sections 3 and 4 while section 5 discussed computation of a hypergeometric-type series. In section 6 empirical data modeling is considered and section 7 concludes. 2. COM-Poisson type Negative Binomial Distribution: Definition and Properties In this section we first introduce a hypergeometric-type series which is to be used in the ensuing sections. The proposed COM-Poisson type negative binomial distribution is then defined and its important distributional properties investigated. 2.1. A hypergeometric-type series In what follows it will be handy to define the series  ( a ) ( a ) ( a ) p k 1 k 2 k m k , m H (a1 , a 2 ,, am ; b; p )    k! (b) k  k 0

where (a) k  a (a  1) (a  k  1) is the Pochhammer’s notation (see Johnson et al., 2005). Obviously, when both  and m are positive integers, it reduces to m F (a1 , a 2 ,  , a m ; b, b, , b; p) , a particular generalized hypergeometric function. With this notation we have 

 ( )

k

p k / (k !) 1 H  1 ( ;1; p)  1F 1 ( ;1,1,,1; p) .

k 0

Some important special cases of 1 H  1 ( ;1; p ) are (i)

1

H 0 ( ;1; p)  (1  p )  .

(ii) 1 H  1 (1;1; p )  Z ( p,   1) . (iii) lim 1H  1 ( ;1; p)  1   p .  



(iv) lim 1H  1 ( ;1; p)   k /(k!)  Z ( ,  ) , when  p   is finite positive.  

k 0

2.2 COM-Poisson type Negative Binomial distribution The generalization of the negative binomial and COM-Poisson distributions is given as follows. Definition 1. A discrete random variable (rv) X is said to follow the COM-Poisson type Negative Binomial distribution (denoted as COM - NB( , p,  ) ) if its pmf is given by P ( X  k )  ( ) k p k /{(k! ) 1 H  1 ( ;1; p)}, k  0, 1, 2, 

(3) 

For any  0, 0  p  1,   1 , it can be checked that the series

 ( )

k

p k / (k !) converges

k 0

for 0  p  1,  1 , since the ratio of successive terms of the series tends to 0 as k   (ratio test of convergence).

3

Remark 1. When   1 , 1 H  1 ( ;1; p ) does not converge and the distribution is undefined. Remark 2. When   1 is an integer the pmf in (3) can be represented by using the generalized hypergeometric function as ( ) k p k (4) P( X  k )  , k  0,1, 2,  (k!) 1F 1 ( ; 1,1, ,1; p ) where 1 F 1 ( ; 1,1, ,1; p ) 



 k 0

( ) k pk ,   0, 0  p  1,   1 . (1) k (1) k  (1) k k !

2.3 Probability recurrence relation and plots of probability mass function The COM-NB pmf has a simple recurrence relation given by P( X  k  1) (  k ) p  , P( X  k ) (k  1)

(5)

1

with P( X  0)   1 H  1 ( ;1; p ) . This is useful for the computation of the probabilities. The computation of 1 H  1 ( ;1; p ) is examined in Section 5. Plots of the COM-NB pmf are given in Figures 1 to 6 to display some features of the distribution. [Figures 1 to 6 here] From the plots of the pmf it is seen that the distribution is either unimodal with nonzero mode(s) or non-increasing with the mode at 0. Also, it is clear that the distribution is under dispersed relative to the negative binomial distribution for   1 . 2.4 Related distributions and Formulation The COM-NB distribution is related to many known distributions with a number of well-known distributions as particular cases. It can be formulated as weighted distributions of known distributions. Also a number of known distributions can be seen as limiting distributions of the COM-NB distribution. Besides, the COM-NB distribution is also a member of well-known families of discrete distributions. 2.4.1 Important particular cases i. Negative binomial distribution (Johnson et al., 2005): For   1 and integer  (  0) the COM - NB( , p,  ) pmf given in (3) reduces to the negative binomial distribution (Johnson et al., 2005) with pmf  k ( ) k pk P( X  k )   p (1  p ) , k  0,1, 2,  k! 1 H 0 ( ;1; p) k        k  1 ( ) k  (  1)  (  k  1)  . where    (1) k     k k! k!  k  k 

4

  k  1  as the “combination with repetition”, while Comtet (1974, Riordan (1958) referred to   k   p. 16) called it the “binomial coefficient with repetition” and used the symbol . k ii. COM-Poisson distribution (Conway and Maxwell, 1962, Shmueli et al., 2005): For  1 , the COM - NB( , p,  ) reduces to the COM-Poisson distribution with parameters p and   1 . iii. Poisson distribution (Johnson et al., 2005): For  1 ,   2 , the COM - NB( , p,  ) reduces to the Poisson distribution with parameter p . iv. Exponentially weighted Poisson distribution: Taking 1 ,   2 and replacing the parameter p by  e , the exponentially weighted Poisson distribution with weight function e K can be seen as a special case of COM - NB( , p,  ) . The resulting pmf is given by P( X  k ) 

e k e  ( ) k  (exp( )1) e  exp( ) ( e ) k . e  k! k!

Remark 3. Since the parameter p satisfies 0  p  1 , the parameter space for the resulting COMPoisson distribution, obtained as a particular case of the COM - NB( , p,  ) , is restricted. 2.4.2 Formulation as weighted distributions The COM - NB( , p,  ) distribution can be formulated as a weighted distribution. For example, consider the NB ( , p) distribution with pmf given by P ( X  k )  ( ) k p k / k!, k  0, 1, 2,  It is easy to see that the weighted NB distribution with weight function 1 /(k !) 1 gives the COM - NB( , p,  ) in (3). Consider the COM-Poisson pmf in equation (1) with the restriction that 0  p  1 . It is easy to check that the weighted COM-Poisson distribution with weight function ( ) k also gives the COM - NB( , p,  ) in (3). Remark 4. As seen in Remark 1 that the pmf of COM-NB distribution is defined only for   1 , it follows that the weight function 1 /(k !) 1 in the weighted NB distribution is always less than or equal to 1. 2.4.3 Limiting distributions i. Bernoulli distribution is a limiting distribution of COM-NB: For given  or p if    , we get the Bernoulli distribution with pmf

P( X  0)  1 /(1   p ) and P( X  1)   p /(1  p) , 5

as the limiting distribution of COM - NB( , p,  ) since lim 1H  1 ( ;1; p )  1   p .  

ii. Binomial distribution is the limiting distribution of the sum of COM-NB rv’s: For given  or p when   , the distribution of the sum of n independent and identically distributed COM - NB( , p,  ) rv’s will tend to the binomial distribution with parameters n and  p /(1  p) (by the arguments of item i above). iii. COM-Poisson is a limiting distribution of COM-NB: When    and  p   , a finite positive number, we get as limiting distribution the COM - Poisson ( ,  ) distribution with parameters  and p . Proof. lim

 

k 1 ( ) k p k 1 (1  1 / ) (1  (k  1) / ) (p )k  lim  (k!) 1 H  1 ( ;1; p) (k!)    ( k !) Z (  ,  ) 1 H  1 ( ;1; p ) (p ) j  Z ( ,  ) .  j  0 ( j!) 

since lim (1  1 / )(1  (k  1) / )  1 and lim 1 H  1 ( ;1; p )    

 

2.4.4 COM-NB distribution as member of families of discrete distributions i. Equation (4) shows that the COM-NB is a member of the generalized hypergeometric family (Kemp, 1956; see Johnson et al. 2005, p. 89, for details) which has pmf of the form (a1 ) k ...(ar )k  k . P( X  k )  k !(b1 )k ...(bs ) k r Fs ( ; a1, , ar ; b1 , , bs ;  ) ii. For known  and  the COM-NB is also a member of the modified power series distribution (Gupta, 1974; 1975). The MPSD has the pmf of the form A( x) g ( p) x , P ( X  x)  f ( p) where x is a non-negative integer, A(x) > 0 and g(p) is a positive, finite and differentiable function of p. In the case of COM - NB( , p,  ) , set g ( p )  p , A( x )  ( ) x / ( x !) and f ( p) 1 H  1 ( ;1; p ) . iii. For fixed  , the COM - NB( , p,  ) is also a member of the exponential family of distribution. To see this, consider the likelihood function for a set of n independent and identically distributed observations x1 , x 2 , , x n from the COM-NB ( , p,  ) given by n

 xi n

L( x1, x2 , , xn  , p,  )  p i 1

n

 ( ) xi / {1 H 1 ( ;1; p )}n  ( xi !)

i 1 n

 xi p

i 1

i 1

n

n

 ln xi !  ln ( ) xi e i 1 e i 1 {1 H  1 ( ;1; p )} n 

6

n

 ln ( ) xi

 p S1 e S 2 e i1 n

n

i 1

i 1

{1 H  1 ( ;1; p )} n

(6)

where S1   x i , S 2   ln xi ! . It is clear that the COM - NB( , p,  ) is a member of the exponential family only when  is known or fixed. When all the parameters are unknown the COM - NB( , p,  ) does not belong to the exponential family just like the NB distribution. Remark 5. The COM-NB ( , p,  ) distribution can be viewed as a continuous bridge in the range of the parameter  ( 1     ) between the negative binomial (   1 ), COM-Poisson (     1,  1 ) and Bernoulli (    ) distributions. 2.5 Cumulative distribution function and probability generating function The cumulative distribution function (cdf) of the COM-NB distribution is seen to be

( )r 1 p r 1 2 H (  r  1;1; r  2; p) P( X  r ) 1  {(r  1) !} 1 1 H 1( ;1; p ) ( ) r 1 p r 1 2 F (  r  1,1; r  2, r  2, , r  2; p ) , {(r  1) !} 1 1 F 1 ( ; 1,1,  , 1; p ) when   1 , is an integer. In particular, for   1 , it reduces to the cdf of the NB distribution (Johnson et al., 2005, p. 218) given by  1

P ( X  r ) 1 

( )r 1 p r 1(1  p) 2 F1 (  r  1,1; r  2; p) . (r  1)!

The probability generating function (pgf) is given in terms of the series 1 H  1 ( ;1; p) as H ( ;1; p s ) P( s )  E ( s X )  1  1 1 H 1 ( ;1; p) F ( ; 1, 1,  ,1; ps )  1  1 , 0  ps 1 1 F 1 ( ; 1,1,  , 1; p ) where the second expression is for the case when   1 is an integer. 2.6 Moments and their recurrence relations Denoting E ( X r )   r' , E ( X [ r ] )  E[ X ( X  1)  ( X  r  1)]   [ r ] and E[{ X  E ( X )}r ]   r , the following recurrence relations for the moments of the COM - NB( , p,  ) distribution can be proved using either the general relations for modified power series distribution (Gupta, 1974; 1975) or by direct manipulation. d  r' 1  p  r'   r' 1' (i) dp d  [ r 1]  p  [ r ]   [ r ]  [1]  r  [ r ] (ii) dp

7

 r 1  p

(iii) Also and

d  r  r r 1  2 dp

d Log[1H 1 ( ;1; p)] dp  d d  d  2  p 1'  p  p Log[1 H  1 ( ;1; p)] dp dp  dp 

1'  p

p

d d2 Log[1 H 1 ( ;1; p )]  p 2 2 Log[1 H  1 ( ;1; p )] . dp dp

Theorem 1. The rth factorial moment E ( X [ r ] )   [ r ] of the COM - NB( , p,  ) is given by

[r ]  

( ) r p r 1 H 1 (  r ; r  1; p ) (r !) 1 1 H 1 ( ;1; p) ( ) r p r 1 F 1 (  r ; r  1, r  1,, r  1; p) , (r !) 1 1 F 1 ( ; 1,1,,1; p )

when   1 , is an integer. Proof: This is easily derived from the definition by finding the rth derivative of the pgf and using the fact that dr H ( ; 1; sp )  ( ) r p r 1H  1 (  r ; r  1; sp ) . r 1  1 ds 2.7 Index of dispersion The index of dispersion (ID), given by ID = variance/mean (Johnson et al., 2005, p.163), is a widely used descriptive index which measures the extent of over (under) dispersion relative to the Poisson distribution. A distribution is under, equi and over dispersed if its ID is less than, equal to and greater than unity respectively. Like the COM-Poisson distribution, the COM - NB( , p,  ) distribution is also able to cater for under, equi and over dispersion in count data: i. For   1 (NB distribution), the distribution is always over dispersed. ii. For   1 (COM-Poisson distribution), the distribution is over (under) dispersed if   () 2 , and for   2 the mean coincides with the variance. iii. For   ()1 , the distribution is under (over) dispersed relative to the COMPoisson distribution. iv. For values of   1 , the distribution is under dispersed relative to the NB distribution. This is verified graphically as well as numerically. 2.7.1 Plots of index of dispersion Contour plots of ID for different choices of parameters are given in Figures 7 to 9. Labels on the lines give value of ID on that line. 8

[Figures 7 to 9 here] From the plots in Figures 7 to 9 it is obvious that the distribution is very flexible with respect to the ID. Interestingly, this family includes a non Poisson distribution with equi-dispersion. Theorem 2. The COM - NB( , p,  ) distribution is equi-dispersed (ID = 1), under-dispersed (ID 1) if E ( X ) , ,  c p , respectively, where c is a constant for fixed values of  ,  with respect to p. Proof: To see this, consider the expressions for the mean and variance in section 2.6. We can

d2

Log[1H 1( ;1; p)] < (>) 0 that is depending on whether dp 2 Log[1H 1 ( ;1; p)] is concave (convex). We find that

derive that ID < (>) 1 if

d2

  d  1 Log [ H (  ; 1 ; p )] = [ H (  ; 1 ; p )] k ( ) k p k 1 / (k !)    1  1 1  1 2 dp  dp k 1  d 1 1 d 1  E( X )  E( X )  2 E( X ) dp p p dp p

Hence d2 1 d 1 E ( X )  () 2 E ( X ) , Log[1 H  1 ( ;1; p )] < (>) 0 implies 2 dp p dp p that is, d 1 Log E ( X )  () or Log E ( X )  () Log p  Log c . dp p We get E ( X )  () cp , where c (> 0) is a constant with respect to p. Clearly, ID = 1, when E ( X )  cp .

d Log[1 H  1 ( ;1; p)] , by using Theorem dp 1, it can be seen that, for constants c (> 0) and b with respect to p , ID = 1 if 1 H  1 ( ;1; p )  exp[c   b] ID < 1 if 1 H  1 ( ;1; p )  exp[c   b  g ( )] , g ( ) < 0 is a decreasing function of  . ID > 1 if 1 H  1 ( ;1; p)  exp[c   b  h( )] , h( ) > 0 is an increasing function of  . Remark 6. Since for the COM - NB( , p,  ) , E ( X )  p

2.8 Distribution of sum and conditional distribution In this section we highlight some important results regarding the sum and conditional distribution given the sum for the COM - NB( , p,  ) distribution. 9

2.8.1 Distribution of sum The sum of n independent COM - NB( , p,  ) forms a continuous bridge between the negative binomial, Poisson and binomial distributions. The sum of COM - NB( , p,  ) variables for i.   1 and p  1 , reduces to the sum of independent negative binomial ( , p ) variables which is a negative binomial (n , p) . ii.   2 and   1 , reduces to the sum of independent Poisson (p) variables which is a Poisson distribution with parameter np . iii. fixed  or p when    , reduces to the sum of independent Bernoulli (p) variables which has a binomial distribution with parameters n and p (1  p ) . 2.8.2 Conditional distribution given the sum Theorem 3. If X ~ COM - NB( x , p,  ) and Y ~ COM - NB( y , p,  ) are independent, then the 

 s  ( x ) j ( y ) s  j conditional distribution of X  j | X  Y  s is proportional to   . ( x   y ) s  j s

Proof: P( X  Y  s )   P( X  j ) P(Y  s  j ) j 0

p s ( x   y ) s



 s  ( x ) j ( y ) s  j      ( s !) 1 H  1 ( x ;1; p)1 H  1 ( y ;1; p ) j 0  j  ( x   y ) s s

Therefore, the conditional distribution of X  j | X  Y  s will be given by 



s  s  ( x ) j ( y ) s  j  s  ( x ) j ( y ) s  j   P( X  j / X  Y  s )     ( x   y ) s j 0  j   j  ( x   y ) s Hence the conditional distribution of X  j | X  Y  s is proportional to



 s  ( x ) j ( y ) s  j   .  j  ( x   y ) s

(7)

Remark 7. a. For  1 , the distribution in (7) reduces to the negative hypergeometric or beta-binomial distribution (Johnson et al., 2005, p. 253). Therefore, (7) can be seen as a COM-Poisson type negative hypergeometric or beta-COM-binomial distribution. b. When  x   y   and  x /( x   y ) is finite and positive the distribution in (7) tends to the COM-Poisson type binomial distribution (Shmueli et al, 2005; Borges et al., 2014) with parameters s and  x ( x   y ) having pmf proportional to 

s     j

 x    y  x

10

   

j

 1   x    x y 

   

s j

c. The COM - NB( , p,  ) can be derived as a limiting form of the distribution in (7). Under the conditions   1, s    , 1 / y  0 and s  / y  p , the pmf in (7) gives the pmf of the COMNB in (3) with parameters ( x , p,  ) . Theorem 4. The COM-Poisson type negative hypergeometric distribution in (7) is a compound COM-Poisson type binomial (Shmueli et al, 2005; Borges et al., 2014) with parameters s and p when p itself is a beta random variable with parameters  x and  y . Proof. The pmf of the compound distribution will be proportional to 1



s   y 1 j s  j  1 dp   j  p (1  p ) p x (1  p )   0   s   ( x  j , y  s  j )  s  ( x ) j ( y ) s  j    =   ,  ( x ,  y )  j  j  ( x   y ) s where  (m, n) is the beta function. 

3. Reliability and stochastic ordering In this section we study the reliability properties and stochastic ordering of the COM - NB( , p,  ) distribution. Stochastic orders have found applications in many diverse areas such as economics, reliability, survival analysis, insurance, finance, actuarial and management sciences (Shaked and Shanthikumar, 2007). For example, Boland, Singh and Cukic (2002) applied stochastic ordering in software debugging to compare two testing methods, namely, partition and random testing, where the comparison of probability of detecting at least one software failure is expressed equivalently in terms of stochastic ordering of random variables. 3.1 Survival function and failure rate function The survival function is given by S (t )  P ( X  t ) 

( )t p t 2 H 1 (  t ; t  1; p ) (t !) 1 H 1 ( ;1; p )

( )t p t 2 F (  t ,1; t  1, t  1, , t  1; p ) , (t !) 1 F 1 ( ; 1,1,  ,1; p) while the failure rate function is 

P( X  t ) 1  P( X  t ) 2 H  (  t ,1; t  1; p) 1  , 2 F (  t ,1; t  1, t  1,  , t  1; p ) where the second expression in terms of hypergeometric function is for the case when   1 is an integer. r (t ) 

11

3.2 Stochastic ordering Definitions: Let X and Y be two discrete random variables with probability mass functions f (x) and g(x). Then X is said to be smaller than Y in the likelihood ratio order denoted by X  lr Y if g(x) = f (x) increases in x over the union of the supports of X and Y; X is smaller than Y in the hazard rate order X  hr Y if rX (t )  rY (t ) for all t; X is smaller than Y in the mean residual life order X  MRL Y if  X (t )  Y (t ) for all t. The following result compares the COM-NB with the NB by using the likelihood ratio order. Theorem 5. The COM-NB is smaller than the NB distribution in the likelihood ratio order i.e. X  lr Y . Proof: If X ~ COM-NB ( , p,  ) and Y ~ NB( , p) , then P(Y  n)  (n!) 1 (1  p ) 1 H  1 ( ; 1; p ) . P ( X  n) This is clearly increasing in n as   1 (Shaked and Shanthikumar, 2007 and Gupta et al., 2012). Hence the result is proved. Corollary 1. Again X  lr Y implies X is smaller than Y in the hazard rate order and subsequently in the mean residual (MRL) life order (Gupta et al., 2012). Symbolically, X  lr Y  X  hr Y  X  MRL Y

4. Log-concavity and its implications 4.1 Log concavity A probability distribution is said to be log concave if its pmf

 fk  ,

f k  0,  k satisfies

f k2  f k 1 f k 1  k . The log concavity of a probability distribution has very important implications on the characteristic of its reliability function, distribution function, failure rate function, tail probabilities and moments. In fact log concavity of a probability distribution is inherited by its reliability function and this produces an increasing failure rate and a monotonically decreasing mean residual life time function (for details see Mark, 1996 and Mark and Bergstrom, 2005). Some important discrete log concave probability distributions are the Bernoulli, binomial, Poisson, geometric and negative binomial distributions. The logarithmic series distribution is an example of a non concave discrete distribution. In this section we deal with the log concavity of the COM–NB and its implications to its reliability properties.

Theorem 6. The COM - NB( , p,  ) has a log-concave probability mass function when  1 . Proof. The COM - NB( , p,  ) pmf is log-concave since for this distribution (Gupta et al., 1997)

  (t ) 

P(t  1) P(t  2) (  t )(t  2)  (  t  1)(t  1)  p > 0. P (t ) P(t  1) (t  1) (t  2) 12

Corollary 2. The following results are the direct consequence of log-concavity (Mark, 1996 and Mark and Bergstrom, 2005): i. COM - NB( , p,  ) has an increasing failure rate function (see Gupta et al., 1997) when  1 . ii. COM - NB( , p,  ) remains log concave if truncated. iii. Convolution of COM - NB( , p,  ) with any other discrete distribution will also result in a log concave distribution. iv. COM - NB( , p,  ) has at most an exponential tail, when  1 that is, lim e a x P( X  x )  0  P ( X  x)  o(e  a x ) for some a  0 as x   . x 

v. For any integer k ,

P( X  i  k ) P( X  j  k ) P( X  i  k )  for i  j , that is, the ratio is P( X  i) P( X  j ) P( X  i )

non increasing in i . 4.2 Strongly unimodality A discrete probability distribution pk  P ( X  k ) with support on the lattice of integers is unimodal if there exists at least one integer M such that pk  pk 1  k  M and pk 1  pk  k  M (Keilson and Gerber, 1971). A discrete probability distribution will be called strongly unimodal if the convolution of it with any other unimodal distribution is again unimodal. Clearly strong unimodality implies unimodality. Some examples of strongly unimodal distributions are the binomial and Poisson distributions (Keilson and Gerber, 1971). Strongly unimodal distributions therefore may have one (unique) or more modes. Theorem 7. The COM - NB( , p,  ) is a strongly unimodal distribution when  1 . Proof. This follows as a consequence of log concavity (see Steutel, 1985, Mark, 1996, Mark and Bergstrom, 2005). Theorem 8.The COM - NB( , p,  ) has (i) a unique mode at X  k if k  /(  k  1)  p  (k  1) /(  k ) , (ii) a non increasing pmf with a unique mode at X  0 if  p  1 , (iii) two modes at X  k and X  k  1 if (  k  1) p  k  . In particular the two modes are at X  0 and X  1 if p  1 . Proof: The results follow easily from the recurrence relation of the probabilities given in (5). Remark 8. A demonstration of the above theorem can be seen in Figure 4, where the pmf with parameters   1, p  0.25 and   1.5 is non increasing with mode at 0 since the parameters satisfy the condition,  p  1 ; in Figure 5, the pmf with parameters   4, p  0.2 and   1.5 has two modes at 0 and 1 as the parameters satisfy the condition p  1 ; in Figure 6 the pmf with 13

parameters   15, p  0.25 and   2 has two modes at 1 and 2 as the parameters satisfy the condition (  1) p  2 . 5. Computation of the infinite series 1 H  1 ( ; 1; p ) For statistical inference and applications it is necessary to compute the normalizing constant 1 H  1 ( ; 1; p ) where  k  . 1 H  1 ( ;1; p )   ( ) k p / ( k !) k 0

Two cases may be considered: Case I. When   1 is an integer it can be regarded as a generalized hypergeometric function. Case II. 1 H  1 ( ; 1; p ) when   1 is not integer the computation may proceed as in Shmueli et al., 2005) To compute 1 H  1 ( ; 1; p ) the following method can be used. Let  1 H  1 ( ;1; p )   a k k 0

k



where ak   k p /  k !  . The ak may be computed with the simple recurrence relation (see Lee et al, 2001)   k  p a with a  1 . ak 1  k 0   k  1 Such two-term recurrence formulae are known to be stable (Ong, 1995; Ong and Muthaloo, 1995), that is, round off errors do not grow relatively quickly with respect to the size of the function. 1 H  1 ( ; 1; p ) is approximated by N 1 H  1 ( ;1; p )   a k k 0

where N is determined by a suitable termination criterion.

6. Empirical data modeling and likelihood ratio test 6.1 Maximum likelihood estimation Suppose that we have a sample of size n from COM-NB with parameters ( , p,  ) reported as grouped frequencies in k classes, that is, ( X, f )  {( x1 , f1 ), ( x2 , f 2, ), ... , ( xk , f k )} , where f i is the k

frequency of the ith observed value xi and n =  f i is the sample size. Then the log-likelihood i 1

function is given by k

k

k



i 1

i 1

i 1

j 0

ln L( x1 , x2 , , xk  , p,  )   f i ln ( ) xi  (  f i xi ) ln p    f i ln xi !  n ln  14

( ) j p j ( j!)

.

The corresponding likelihood equations are  k d f d pj d ln L  0   i ( ) xi  n  ( ) j  i 1 ( ) x d d ( j ! ) d  j  0 i  ( ) j p j 1 k d j ln L  0  (  f i xi ) / p  n  i  1 dp ( j!) j 0



 j 0



 j 0

( ) j p j ( j!)

( ) j p j ( j!)

0

0

  ( ) p j k d j ln L  0   f i ln xi !  n  ( ) j p j ln( j!) /( j!)  0  i 1 d j 0 j 0 ( j!) The analytical solutions of the above likelihood equations are not tractable. Therefore, numerical optimization method is used to obtain the maximum likelihood estimates by maximizing the log likelihood function for the data fitting experiments and the likelihood ratio test.

6.2 Likelihood ratio test The COM - NB( , p,  ) distribution reduces to the NB distribution with parameters ( , p ) when  1 . Since the two distributions are nested we have employed the likelihood ratio criterion to test the following hypotheses: H 0 :  1 , that is, the sample is from NB with parameters  , p H1 :  1 , that is, the sample is from COM-NB with parameters , p,  . Writing β  ( , p,  ) the likelihood ratio test statistic is given by L(βˆ * ; x) LR =  2 ln , L(βˆ , x) where βˆ * is the restricted ML estimates under the null hypothesis H 0 and βˆ is the unrestricted ML estimates under the alternative hypothesis H 1 . Under the null hypothesis H 0 , LR follows a chi-square distribution with one degree of freedom (df). At 5% level of significance the two sided critical region for this test is given by {LR : LR  0.00098  LR  5.02} (for critical values for  2 distribution see Table 5 of Kanji, 2006). Thus there will no evidence against the null hypothesis if 0.00098  LR  5.02 , otherwise H 0 will be rejected. 6.3 A numerical example The frequency data of the number of spots(x) in southern pine beetle, Dentroctonus frontails Zimmerman, (Coleopetra: Scolytidae), in Southeast Texas (Lin, 1985, Ong et al., 2012) considered here is over dispersed with index of dispersion 4.1321. The COM-NB model has been fitted and compared with the negative binomial (NB), COM-Poisson and Generalized Poisson distributions (Consul, 1989). The performances of various distributions are compared using the  2 goodness of fit statistic, the Akaike Information Criterion (AIC). AIC is defined as AIC = -2 ln L + 2k, where k is the number of parameter(s) and ln L is the maximum of log-likelihood for a given data set (Burnham and Anderson, 2004). The likelihood ratio test is performed to compare the COM-NB with NB distribution. The likelihood ratio test rejects H 0 at 5% level. From the results from the data fitting furnished in Table 1, it is seen that the COM-NB may serve as a better empirical model than the NB, COM-Poisson, and generalized Poisson 15

distributions in terms of log-likelihood,  2 and AIC values. Based on the difference in AIC values given in Burnham and Anderson (2002, p. 70), there is considerable empirical support for the COM-NB when compared with the NB COM-Poisson, and generalized Poisson distributions. [Table 1 here] 7. Conclusion A generalization of the negative binomial distribution has been proposed in analogy with the COM-Poisson distribution. Many well-known distributions are particular cases of this COMPoisson type negative binomial distribution while the binomial and COM-Poisson distributions are obtained as limiting distributions. The parameter  of the COM-Poisson type negative binomial distribution adds to its flexibility by enabling us to fit distributions that are not exactly negative binomial or COM-Poisson or Bernoulli, but lie on a continuum (in the range of  ) between this distributions. The proposed distribution belongs to the modified power series and generalized hypergeometric families, and also arises as weighted negative binomial and COMPoisson distributions. Like the COM-Poisson and generalized Poisson distributions, this COMPoisson type negative binomial distribution is also able to model under, equi- and over dispersion. It is shown that the COM-Poisson type negative binomial distribution has many interesting probabilistic properties and is a better empirical model than the COM-Poisson and generalized Poisson distributions. Acknowledgment: The authors would like to specially acknowledge the comments and suggestions of the Associate Editor and referees which have improved the paper considerably. The authors also wish to acknowledge support in parts from the Ministry of Education FRGS grant FP010-2013A and University of Malaya's UMRGS grant RP009A-13AFR.

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