Decision Science Letters

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1 Decision Science Letters 8 (09) 37 3 Contents lists vilble t GrowingScience Decision Science Letters homepge: The negtive binomil-weighted Lindley distribution Sunthree Denthet * nd Prmoch Promin College of Industril Technology King Mongkut s University of Technology North, Thilnd C H R O N I C L E A B S T R A C T Article history: Received October 9, 08 Received in revised formt: October 0, 08 Accepted November, 08 Avilble online November, 08 Keywords: Count dt nlysis Mied negtive binomil distribution Weighted Lindley distribution This pper proposes new distribution nmed the negtive binomil-weighted Lindley. The study uses the mimum likelihood estimtion to estimte the prmeters of the proposed distribution nd compres the performnce of the new method with other distributions. The study finds tht the negtive binomil-weighted Lindley distribution, obtined by the miing the negtive binomil distribution with the weighted Lindley distribution is nother mied negtive binomil distribution nd my provide n pproprite fit for dt estimtion with overdispersion. Some chrcteristics of the proposed distribution, such s men nd vrince re lso derived. 08 by the uthors; licensee Growing Science, Cnd.. Introduction A count dt distribution is only non-negtive integers in its domin. We typiclly use the count dt distribution to model the number of occurrences of certin event. The Poisson nd negtive binomil (NB) distributions re the count dt distribution for emples. The stndrd distribution for modeling count dt hs been the Poisson distribution, which is proper model for counting the number of occurrences over time intervl t rndom when not mny occurrences re observed within short period of time. They occur t constnt rte through time, nd one occurrence of the phenomenon does not lter the probbility of ny future occurrence (Riner, 008; Tem, 05). Let X ~ Poisson( ) be Poisson distributed with prmeter. The probbility mss function (pmf) of X is given by ep( ) () f( ), 0,,,..., 0.! Then, the men nd vrince re given by EX ( ) nd Vr( X ). Equlity of men nd vrince, clled equl dispersion, is clssic chrcteristic of the Poisson distribution. Moreover, there re other ctegories of dispersion which re overdisperssion when the vrince is greter thn the men nd underdispersion where the vrince is smller thn the men (Hight, 967). The NB distribution is * Corresponding uthor. E-mil ddress: srd_kmutnb@hotmil.com (S. Denthet) 09 by the uthors; licensee Growing Science, Cnd. doi: 0.567/.dsl

2 38 populr lterntive distribution for modelling overdispersed count dt becuse it is more fleible in ccommodting overdispersion in comprison with the Poisson model. The NB distribution is miture of Poisson distribution by miing the Poisson nd gmm distribution. Applictions using the NB distribution cn be found in mny res, for instnce, economics, ccident sttistics, biosttistics nd cturil science. The problem of overdispersion is usully solved by introducing mied NB distribution. In severl studies, it is shown tht mied NB distribution provides better fit on count dt compred with the Poisson nd the NB distribution. These include the Poisson-inverse Gussin (Klugmn et l., 008), negtive binomil-inverse Gussin (Gómez-Déniz et l., 008), negtive binomil-lindley (Zmni & Ismil, 00), negtive binomil-bet Eponentil (Pudprommrt et l., 0), nd negtive binomil-erlng (Kongrod et l., 04). The Lindley distribution hs been generlized by mny reserchers in recent yers. The Lindley distribution is the miture of eponentil ( ) nd Gmm (, ) distributions (Lindley, 958). Subsequently Ghitny et l. (008) investigted Lindley distribution in the contet of relibility nlysis. Subsequently, weighted Lindley (WL) distribution is proposed for modelling survivl dt. A rndom vrible X follows the WL distribution with prmeters 0 nd nd the probbility density function (pdf) is follows, (- ) f( ) = ( + )ep( -( -)), for > 0. ( - + ) Let X ~WL(, ), then its moment generting function (mgf) of X is given by ( )( t ) M X () t = ( - + )( - - t) Some plots of the WL pdf with some specified vlues of nd re shown in Fig.. () (3) Fig.. Some pdf plots of the WL distribution In this reserch, count distribution, which is represented s n lterntive distribution for overdispersed count dt, nmely the negtive binomil- weighted Lindley (NB-WL) distribution is developed. The NB-WL distribution is miture of the NB nd WL distributions. The method is more fleible lterntive to the Poisson nd NB distribution. Some of the chrcteristics of the proposed distribution cn be studied through fctoril moments, e.g., men nd vrince. The prmeters of the proposed distributions re estimted by using the mimum likelihood estimtion (MLE). The MLE is populr technique for estimting prmeter of given function which mkes tht likelihood function mimum nd it is lso powerful nd unbised estimtion in estimting prmeters (Hmid, 04). The proposed distribution is compred with the performnce of Poisson nd NB distributions.. Methodology. Reserch obectives The obectives of this reserch re to propose new mied distributions, to derive the prmeter estimtion of the proposed distributions by using the MLE method nd compres the efficiencies of the proposed distribution with other distributions for count dt nlysis.

3 . The mterils S. Denthet nd P. Promin / Decision Science Letters 8 (09) 39 The mterils of this reserch re s high performnce personl computer for running the coded progrm. The mimum likelihood estimtes rˆ, ˆ nd ˆ for the prmeters r, nd respectively, re tken by solving itertively differentil equtions to zero. These differentil equtions re not in closed form nd numericl method cn be employed to obtin the epecttions of them. The MLE solution of rˆ, ˆ nd ˆ cn be obtined by solving the resulting equtions simultneously using optim function in R lnguge..3 The methods The methods of the reserch re to investigte pmf nd some properties of the NB-WL distribution. To estimte the prmeters of the NB-WL distribution, MLE method is implemented. Rndom vrite genertion of the NB-WL distribution is derived nd ppliction of the NB-WL distribution to rel dt set hs been studied by compring with the Poisson nd NB distributions using the Kolmogorov- Smirnov (K-S) from the dgof pckge of R lnguge (Arnold & Emerson, 0). 3. Results This section presents the results of the reserch nd provides the probbility mss function (pmf) of the proposed distribution. Moreover, some chrcteristics including the plots of the pmf with vrious vlues of prmeters, prmeter estimtion, rndom vrite genertion, nd ppliction of the proposed distribution to rel dtset re included in ech prt. 3. The propose Distribution We propose new mied NB distribution which is n NB-WL distribution obtined by miing the NB distribution with WL distribution. The distribution hs three prmeters, nmely, r, nd. We begin with generl definition of the NB-WL distribution which will consequently revel its the probbility mss function (pmf). Fig.. displys the NB-WL pmf plots with some specified prmeter vlues of r, nd. Definition. Let X be rndom vrible following NB distribution with prmeters r nd p ep( ), X ~ NB ( r, p ep( )). If is distributed s the WL distribution with positive prmeters nd, denoted by ~ WL(, ), then X is clled NB-WL rndom vrible. Theorem. Let X ~NB-WL( r,, ). The pmf of X is given by ( )( ) f( ; r,, ) ( ), 0,,,... ( )( 0 (4) where 0 nd. Proof. If X ~ NB( r, p ep( )) nd ~ WL(, ), then the pmf of X cn be obtined by where f ( ) is epress s f ( ) f ( ) g( ;, ) d, 0 f( ) ep( r)( ep( )) ( ) ep( ( r ). 0 (5)

4 30 By substituting f ( ) into f ( ) f( ) g( ;, ) d, thus 0 f ( ) ( ) ep( ( r ) g( ;, ) d ( ) M ( ( r ) (6) Substituting M ( ( r ) the mgf of the WL distribution in the eqution bove, the pmf of the NB-WL ( r,, ) is given s ( )( ) f( ; r,, ) ( ), ( )( 0 (7) Fig.. The pmf of the NB-WL distribution of some specified vlues of r, nd 3. Chrcteristics of the NB-WL distribution Some chrcteristics of the NB-WL distribution will be discussed s follows. The fctoril moment of the NB-WL distribution is introduced. Some of the most importnt structures nd chrcteristics of the NB-WL distribution cn be studied through fctoril moments. Theorem. If X ~NB-WL( r,, ). the fctoril moment of order of X is ( r) ( )( ) ( X) ( ), 0,,,... () r 0 ( )( for 0 nd. (8) Proof. Gómez-Déniz et l. (008) showed tht the fctoril moment of order of mied NB distribution cn be epressed in the terms of elementry function by ( r) (ep( )) ( r) ( X) E E(ep( ) ). () r ep( ) () r (9)

5 S. Denthet nd P. Promin / Decision Science Letters 8 (09) 3 Using the binomil epnsion of (ep( ) ), then ( X ) cn be written s ( r) ( r) ( X ) ( ) E(ep( ( )) ( ) M(. () r 0 () r 0 X From the mgf of the NWL distribution with t =, the ( ) is finlly given s ( r) ( )( ) ( X ) ( ). () r 0 ( )( Definition. Let X ~NB-WL( r,, ). some properties of X re s follows ) The first two moments bout zero of X re EX ( ) r( -), () E( X ) r( r) - r(r) r, (3) ) The men nd vrince of X respectively, re EX ( ) r( -), (4) Vr( X ) r( r ) - r ( r ). (5) ( )( k ) where k. ( )( k) (0) () 3.3 Applictions study of NB-WL distribution We illustrted the NB-WL, NB nd Poisson distributions by pplying the number of hospitlized ptients with dibetes t Rtchburi hospitl, Thilnd. The log-likelihood vlues nd the p-vlues of K-S test for the discrete goodness of fit test re summrized in Tble. The epected frequencies of the NB-WL distribution re close to the observed frequencies, the vlues of K-S test of NB-WL distribution is smller thn the vlues of the K-S test of the Poisson nd NB distributions nd Also, bsed on the p-vlues of K-S test, the proposed distribution is pproprite to fit the dt compred to the Poisson nd NB distributions. Tble Observed nd epected frequencies for number of hospitlized ptients with dibetes No. of No. of Epected vlue by fitting distribution hospitliztion cses Poisson NB NB-WL Totl Prmeter estimtes ˆ.7 rˆ 4.07 rˆ 4.5 pˆ 0.48 ˆ 0.5 ˆ.0 log-likelihood K-S test p-vlue <

6 3 4. Conclusions This work hs proposed new mied negtive binomil distribution clled the negtive binomil-new weighted Lindley distribution. In prticulr, some of the most importnt chrcteristics of the distribution cn be studied through fctoril moments, e.g., men, vrince, skewness, nd kurtosis. In the ppliction of the NB-WL distribution, we hve compred the ccurcy of the proposed distribution with the Poisson nd NB distributions. The usefulness of the NB-WL distribution hs been illustrted by the number of hospitlized ptients with dibetes t Rtchburi hospitl, Thilnd. We hve used the log-likelihood nd p-vlues of the K-S test for the goodness of fit for model selection purpose. Finlly, the result of this study hs shown tht the NB-WL distribution provides better fit compred with the Poisson nd NB distributions. Obviously, the NB-WL distribution is n lterntive distribution to the other for count dt. Acknowledgement The uthors would lso like to thnk College of Industril Technology King Mongkut s University of Technology North Bngkok their finncil support during my study. References Arnold, T. B., & Emerson, J. W. (0). Nonprmetric Goodness-of-Fit Tests for Discrete Null Distributions. R Journl, 3(), Ghitny, M. E., Atieh, B., & Ndrh, S. (008). Lindley distribution nd its ppliction. Mthemtics nd Computers in Simultion, 78(4), Gómez-Déniz, E., Srbi, J. M., & Clderín-Oed, E. (008). Univrite nd multivrite versions of the negtive binomil-inverse Gussin distributions with pplictions. Insurnce: Mthemtics nd Economics, 4(), Hmid, H. (04). Integrted Smoothed Loction Model nd Dt Reduction Approches for Multi Vribles Clssifiction (Unpublished doctorl disserttion). Universiti Utr Mlysi, Kedh, Mlysi. Hight, F. (967). Hndbook of the Poisson distribution. John Wiley nd Sons, New York. Klugmn,S., Pner, H. nd Willmot, G. (008). Loss models: from dt to decisions. 3rd. John Wiley nd Sons. Kongrod, S., Bodhisuwn, W., & Pykkpong, P. (04). The negtive binomil-erlng distribution with pplictions. Interntionl Journl of Pure nd Applied Mthemtics, 9(3), Lindley, D. V. (958). Fiducil distributions nd Byes' theorem. Journl of the Royl Sttisticl Society. Series B (Methodologicl), 0(), Pudprommrt, C., Bodhisuwn, W., & Zeephongsekul, P. (0). A new mied negtive binomil distribution. Journl of Applied Sciences(Fislbd), (7), Riner,W. (008). Econometric nlysis of count dt. Librry of congress control, New York. Tem, R.C. (05). R: A Lnguge nd Environment for Sttisticl Computing. R Foundtion for Sttisticl Computing. Vienn, Austri. Zmni, H., & Ismil, N. (00). Negtive binomil-lindley distribution nd its ppliction. Journl of Mthemtics nd Sttistics, 6(), by the uthors; licensee Growing Science, Cnd. This is n open ccess rticle distributed under the terms nd conditions of the Cretive Commons Attribution (CC-BY) license (