THE NEGATIVE BINOMIAL-ERLANG DISTRIBUTION WITH APPLICATIONS. Kasetsart University Chatuchak, Bangkok, 10900, THAILAND

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1 International Journal of Pure and Applied Mathematis Volume 92 No , ISSN: (printed version; ISSN: (on-line version url: doi: PAipam.eu THE NEGATIVE BINOMIAL-ERLANG DISTRIBUTION WITH APPLICATIONS Siriporn Kongrod 1, Winai Bodhisuwan 2, Prasit Payakkapong 3 1,2,3 Department of Statistis Kasetsart University Chatuhak, Bangkok, 10900, THAILAND Abstrat: This paper introdues a new three-parameter of the mied negative binomial distribution whih is alled the negative binomial-erlang distribution. This distribution obtained by miing the negative binomial distribution with the Erlang distribution. The negative binomial-erlang distribution an be used to desribe ount data with a large number of zeros. The negative binomialeponential is presented as speial ases of the negative binomial-erlang distribution. In addition, we present some properties of the negative binomial-erlang distribution inluding fatorial moments, mean, variane, skewness and kurtosis. The parameter estimation for negative binomial-erlang distribution by the maimum likelihood estimation are provided. Appliations of the the negative binomial-erlang distribution are arried out two real ount data sets. The result shown that the negative binomial-erlang distribution is better than fit when ompared the Poisson and negative binomial distributions. AMS Subet Classifiation: 60E05 Key Words: mied distribution, ount data, Erlang distribution, the negative binomial-erlang distribution, overdispersion Reeived: Deember 4, 2013 Correspondene author 2014 Aademi Publiations, Ltd. url:

2 390 S. Kongrod, W. Bodhisuwan, P. Payakkapong 1. Introdution The Poisson distribution is a disrete probability distribution for the ounts of events that our randomly in a given interval of time, sine data in multiple researh fields often ahieve the Poisson distribution. However, ount data often ehibit overdispersion, with a variane larger than the mean. The negative binomial (NB distribution is employed as a funtional from that whih relaes the overdispersion restrition of the Poisson distribution. The NB distribution is a miture of Poisson distribution by miing the Poisson distribution and gamma distribution. The NB distribution has beome inreasingly popular as a more fleible alternative to ount data with overdispersion. The NB distribution was first introdued by Pasal [2]. Montmort [13] applied the NB distribution to represent the number of times that a oin should be flipped to obtain a ertain number of heads. Student [16] used the NB distribution as an alternative to the Poisson distribution. Eggenberger and Polya [5] has been shown to be the limiting form of the NB distribution. Ever sine, there has been an inreasing number of appliations of the NB distribution have been developed in a parallel way. For instant, the Poisson-inverse Gaussian distribution [15], [9], the negative binomial-inverse Gaussian distributions [4], [15] and the negative binomial-beta Eponential distribution [3]. In this paper we introdue a new mied negative binomial distribution, whih is alled the negative binomial-erlang (NB-EL distribution. The NB- EL distribution an be used to desribe ount data distribution with a large number of zeros in Poisson distribution. The Erlang (EL distribution was introdued by Agner Erlang [1] was the first author to etend the eponential distribution with his method of stages. He defined a non-negative random variable as the time taken to move through a fied number of stages, spending an eponential amount of time with a fied rate in eah one. The NB-EL distribution is obtained by miing the NB distribution whose probability of suess parameter p follows the EL distribution. In other words, if the probability of suess parameter p of the NB distribution has the EL distribution, then the resulting distribution is referred to as the NB-EL with parameters r, k and. The negative binomial-eponential has three-parameter is speial ase of the NB-EL distribution. In partiular, various strutural properties of the new distribution are derived, inluding epansions for its fatorial moments, mean, variane, skewness and kurtosis.

3 THE NEGATIVE BINOMIAL-ERLANG The Negative Binomial Distribution The negative binomial distribution is a natural and more fleible etension of Poisson distribution and allows for overdispersion relative to the Poisson distribution. If X is the NB distribution with parameter r > 0 and 0 < p < 1 then we an write the pmf as ( r + 1 f(;r,p = p r (1 p ; = 0,1,2,... (1 Note that p is a probability of suess, the eperiment is repeated as many times as required to obtain r suesses. The first two moments about zero and the fatorial moment of order m of the NB distribution are given respetively E(X = r(1 p, p Var = r(1 p p 2, µ [m] (X = Γ(r +m] Γ(r (1 pm p m, m = 1,2,... (2 The log likelihood funtion of the NB distribution is given by L(r,p = n [( r +i 1 i p r (1 p i ], then, we an write the log likelihood funtion of the NB(r,p as logl(r,p = n ( r +i 1 log +nrlog(p i ( n log(1 p i. (3

4 392 S. Kongrod, W. Bodhisuwan, P. Payakkapong 3. The Erlang Distribution The EL distribution was introdued by Agner Erlang [1]. It is a ontinuous probability distribution with wide appliability mainly due to its relative to the eponential distributions. The probability density funtion (pdf of the EL distribution is given by g( = k k 1 e, (4 (k 1! for > 0 and k, > 0. Note that parameter k is alled the shape parameter and the parameter is alled the rate parameter. The mean of the EL distribution is given as The variane of the EL distribution is then given as E(X = k. (5 Var(X = k 2. (6 The moment generating funtion of the EL distribution is defined by M X (t = ( 1 t k for t > 0. (7 4. The Negative Binomial-Erlang Distribution Definition 1. Let X be a random variable of the NB-EL (r,k, distribution, when the NB distribution have parameters r > 0 and p = ep( λ, where λ is distributed as the EL distribution with positive parameters k and, i.e., X λ NB(r,p = ep( λ and λ EL(k,. Theorem 2. Let X NB-EL(r,k,. The pmf of X is given by ( r+ 1 f(;r,k, = for > 0 and k, > 0. ( ( ( 1 +(r + k, (8

5 THE NEGATIVE BINOMIAL-ERLANG Proof. If X λ NB(r,p = ep( λ in Eq. 1 and λ EL(k, in Eq. 4, then the pmf of X an be obtained by f(;r,k, = where, f 1 ( λ is defined by ( r + 1 f 1 ( λ = ( r + 1 = 0 f 1 ( λg(λ;k,dλ (9 e λ (1 e λ By substituting Eq. 10 into Eq. 9, we obtain ( ( ( r + 1 f( λ = ( 1 0 ( ( r + 1 = ( ( 1 e λ(r+. (10 e λ(r+ f 2 (λ;α,βdλ ( 1 (M λ ( (r +. (11 Substituting the moment generating funtion of EL distribution Eq. 7 into Eq. 11, the pmf of NB-EL (r,k, is finally given as ( ( ( r+ 1 f(;r,k, = ( 1 k. +(r + Many well known distributions are subsumed by the NB-EL distribution. Net, we display graphs of the pmf of the NB distribution with various values of parameters are shown in Figure 1. Corollary 3. Ifk = 1thentheNB-EL distributionreduestothenegative binomial-eponential (NB-E distribution with pmf given by ( ( ( r + 1 +(r + f(;r, = ( 1, (12 where = 0,1,2,... for r and > 0. Proof. If X λ NB(r,p = ep λ and λ EL(k = 1,, then the pmf of X is ( ( ( r + 1 +(r + f(;r, = ( 1

6 394 S. Kongrod, W. Bodhisuwan, P. Payakkapong (a (b Probability Probability ( (d Probability Probability (e (f Probability Probability Figure 1: (a-f: The probability mass funtion of the NB-EL distribution of some values of parameters: (a r=0.5, k=0.5, =0.5, (b r=0.5, k=1, =0.5, ( r=0.5, k=2, =0.5, (dr=5, k=1, =1.5, (e r=5, k=1, =2 and (f r=5, k=2, =5

7 THE NEGATIVE BINOMIAL-ERLANG ( ( ( r + 1 +(r + = ( 1. From Corollary 3, we find the negative binomial-eponential distribution displayed in Eq. 12, whih introdued by Panger and Willmot [17]. 5. Properties of the NB-EL Distribution In this setion, we have studied the properties of the NB-EL distribution, whih inludes the fatorial moments, mean, variane, skewness and kurtosis are given as follow Theorem 4. If X NB-EL(r, k,, then the fatorial moment of order m of X is given by µ [m] (X = Γ(r +m Γ(r m where = 0,1,2,... for r,k and > 0. ( ( m ( 1 k, (13 (m Proof. If X λ NB(r,p = ep( λ in Eq. 1 and λ EL(k, in Eq. 4, then the fatorial moment of order m of X an be obtained by µ [k] (X = E λ [µ k (X λ]. The fatorial moment of order m of the NB distribution in Eq. 2, µ [m] (X beomes ( Γ(r +m (1 e λ m Γ(r +m µ [m] (X = E λ Γ(r e λm = E λ (e λ 1 m. Γ(r Using a binomial epansion of (e λ 1 m, then shows that µ [m] (X an be written as µ [m] (X = = Γ(r +m Γ(r Γ(r +m Γ(r m ( m ( 1 E λ (e λ(m m ( m ( 1 M λ (m.

8 396 S. Kongrod, W. Bodhisuwan, P. Payakkapong From the moment generating funtion of the EL distribution in Eq. 7, we have finally that µ [m] (X an be written as µ [m] (X = Γ(r +m Γ(r m ( ( m ( 1 k. (m From the fatorial moments of the NB-EL distribution, it is straightforward to redue the first four moments given in Eq Eq. 17, variane in Eq. 18, skewness in Eq. 19 and kurtosis in Eq. 20. E(X = r(ϕ 1 1, (14 E(X 2 = (r 2 +rϕ 2 (2r 2 +rϕ 1 +r 2, (15 E(X 3 = (r 3 +3r 2 +2rϕ 3 (3r 3 +6r 2 +3rϕ 2 +(3r 3 +3r 2 +rϕ 1 r 3, (16 E(X 4 = (r 4 +6r 3 +11r 2 +6rϕ 4 (4r 4 +18r 3 +26r 2 +12rϕ 3 +(6r 4 +18r 3 +19r 2 +7rϕ 2 (4r 4 +6r 3 +4r 2 +rϕ 1 +r 4, (17 Var(X = E(X 2 (E(X 2 = (r 2 +rϕ 2 rϕ 1 (1+rϕ 1, (18 Skewness(X = [ E(X 3 3E(X 2 E(X+2[E(X] 3 / ] σ 3 [ = (r 3 +3r 2 +2rϕ 3 (3r 2 +3rϕ 2 +rϕ 1 +3r 2 (ϕ 2 2 (3r 3 +3r 2 ϕ 1 ϕ 2 +2r 3 (ϕ 2 3] /σ 3, (19 Kurtosis(X = = [ E(X 4 4E(X 3 E(X+6E(X 2 (E(X 2 3[E(X] 4 / ] σ 4 [ (r 4 +6r 3 +11r 2 +6rϕ 4 (6r 3 +18r 2 +12rϕ 3 3r 4 (ϕ 1 4 +(7r 2 +7rϕ 2 4rϕ 1 6r 3 (ϕ 1 3 +(12r 3 +12r 2 ϕ 1 ϕ 2 (4r 4 +12r 3 +8r 2 ϕ 1 ϕ 3 +(6r 4 +6r 3 (ϕ 1 2 ϕ 2 ]/σ 4, (20

9 THE NEGATIVE BINOMIAL-ERLANG where ( i k ϕ i =. 6. Parameters Estimation The estimation of parameters for NB-EL distribution via the maimum-likelihood estimation (MLE method proedure will be disussed. The likelihood funtion of the NB-EL(r,k, is given by n i L(r,k, = ( r+i 1 i ( ( i ( 1 with orresponding log-likelihood funtion n ( r+i 1 L = logl(r,k, = log + n i ( i log ( 1 ( i +(r + k. (21 k. (22 +(r + The first order onditions for finding the optimal values of the parameters obtained by differentiating Eq. 22 with respet to r,k and give rise to the following differential equations and L r = + L n k = L r n i 1 k=0 n i = log(r +k k i i ( i [ ( 1 i n i ( i ( 1 k [ ( 1 ( i +(r+ ( i ( 1 i [ (+(r+ ] k+1 k +(r+ ] klog [ +(r+ +(r+ ] k ( i [ ( 1 k k 1 (r+ [ ] k ( 1 ( i (+r+ k+1 ] +(r+, (23 ], (24. (25

10 398 S. Kongrod, W. Bodhisuwan, P. Payakkapong We estimated parameters of the NB-EL distribution with equating Eq Eq. 25 to zero, the MLE solutions of ˆr,ˆk and ĉ an be obtained by solving the resulting equations simultaneously using a numerial proedure, the Newton- Raphson method. 7. Illustrative Eamples The NB-EL distribution is applied on two real ount data sets. The details are presented in the following eamples. Eample 1. We use a real data set is numbers of inured from the aident on maor road in Bangkok in 2007 [3]. The data was olleted by Department of Highways, Ministry of Transport, Thailand. We use the real data are fitted by the Poisson, NB and NB-EL distributions (Table 1, whih show the observed and epeted frequenies, grouped in lasses of epeted frequeny greater than five for the hi-square goodness of fit test. Based on the p-value, the maimum likelihood method provides very poor fit for the Poisson and the NB distribution. We an see that NB-EL distribution provides the highest p-value of fitting for this data set. Table 1: Observed and epeted frequenies for the aident data Number of Observed Fitting distribution inured frequeny Poisson NB NB-EL } Estimated parameters ˆλ= ˆr = ˆr = ˆp = ˆk= ĉ= Chi-squares Degrees of freedom p-value < Eample 2. We used a data set whih was obtained from Klugman et

11 THE NEGATIVE BINOMIAL-ERLANG al. [15], provides the data set of 9,461 automobile insurane poliies where by the number of aidents of eah poliy has been reorded. We apply this data set to fit with the Poisson, NB and NB-EL distributions. Obtained results (Table 2, shows that the observed and epeted frequenies, grouped in lasses of epeted frequeny greater than five for the hi-square goodness of fit test. The maimum likelihood method provides very poor fit for the Poisson and the NB distribution. Based on p-value we found that the NB-EL distribution seen to be best fit among there three distributions, with p-value Table 2: Observed and epeted frequenies for the aident data Number of Observed Fitting distribution laims frequeny Poisson NB NB-EL Estimated parameters ˆλ= ˆr = ˆr = ˆp = ˆk= ĉ= Chi-squares Degrees of freedom p-value < Conlusion In this paper, we introdue a new three-parameter negative binomial-erlang distribution, NB-EL(r, k,. This distribution obtained by miing the NB with the EL distribution (when the NB distribution have parameters r > 0 and p = ep( λ, where λ is distributed as the EL distribution with positive parameters k and. We showed that the negative binomial-eponential distributions isaspeial ase ofthenb-el distribution. Inaddition, themoments of thenb- EL distribution whih inludes the fatorial moments, mean, variane, skewness

12 400 S. Kongrod, W. Bodhisuwan, P. Payakkapong and kurtosis are derived. Moreover, the parameter estimation of the NB-EL using the maimum likelihood estimation are developed. We inlude an appliation of the NB-EL distribution to fit two real data sets. We found that this data set is best fit with NB-EL distribution among there three distributions. We hope that NB-EL distribution may attrat wider appliations in analyzing ount data. Aknowledgments We would like to thank Dr.Chookait Pudprommarat, who has given many useful omments and valuable suggestions. This work was supported by researh funding from the Researh Professional Development Proet under the Siene Ahievement Sholarship of Thailand (SAST. Referenes [1] A.K. Erlang, Solution of some problems in the theory of probabilities of signifiane in automati telephone ehanges, Elektrotkeknikeren, 13(1917, 513. [2] Blaise Pasal, Varia opera mathematia D.Petri de Fermat, Tolossae, (1679. [3] C. Pudprommarat, W. Bodhisuwan, A new mied negative binomial distribution, Journal of Applied Sienes, 12(2012, , doi: /as [4] E. Gomez-Deniz, J.M. Sarabia and E. Calderin-Oeda, Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with appliations, Insurane: Mathematis and Eonomis, 4(2008, doi: /J.insmatheo [5] F. Eggenberger and G.Polya, Uber die Statistik verketetter Votgange, Zeitshrift fur Angewandte Mathematik and Mehanik, 1 (1923, , doi: /zamm [6] G.E. Willmot, The Poisson-inverse Gaussian distribution as an alternative to the negative binomial, Sandinavian Atuar, 2 (1987, , doi: /

13 THE NEGATIVE BINOMIAL-ERLANG [7] H. Ranger and E. Willmot, Finite sum evaluation of the negative binomial eponential model, Astin Bulletin, 12(1981, [8] J. Lemaire, How to define a bonus-malus system with an eponential utility funtion. ASTIN Bull. 10(1979, [9] L. Tremblay, Using the Poisson inverse Gaussian in bonus-malus systems, ASTIN Bull, 22(1992, [10] L.J. Simon, Fitting negative binomial distributions by the method of maimum likelihood, Pro. Casual. Atuar. So., 48(1961, [11] M. Greenwood and G.U. Yule, An inquiry into the Nature of frequeny distributions representative of multiple happenings with partiular referene to the ourrene of multiple attaks of disease or of repeated aidents, Journal of the Royal Statistial Soiety, 83(1920, [12] M.D. Bressoud, A radial approah to real analysis, Cambridge University Press, Cambridge (2005. [13] P.R. de Montmort, Essay d Analyse sur les Jeu de Hazard, Seonde Edition, Revue Augmentee de plusieurs Lettre, Quillau, Paris (1980. [14] S. Meng,Y. Wei and G.A. Whitmore, Aounting for individual overdispersion in a bonus-malus system, ASTIN Bull.(1999. [15] S.A. Klugman. H.H. Paner and G.E.Willmot, Loss Models: From Data to Deision, John Wiley and Sons USA, (2008. [16] W.S. Gosset Student, On the error of ounting with a Haemaytometer, Biometrika, 5(1907, [17] H.H. Paner and G.E. Willmot, Finite sum eqvluation of the negative binomail-eponential model, Astin Bulletin, 12(1981,

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