The Poisson-generalised Lindley distribution and its applications

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1 Sogklaakar J. Sc. Techol. 38 (6), , Nov. - Dec Orgal Artcle The Posso-geeralsed Ldle dstrbuto ad ts applcatos Weerrada Wogr ad Wa Bodhsuwa* Departmet of Statstcs, Facult of Scece, Kasetsart Uverst, Chatuchak, Bagkok, 19 Thalad. Receved: 3 October 15; Accepted: 5 Februar 16 Abstract The Posso dstrbuto plas mportat role cout data aalss. However, the Posso dstrbuto caot model some data wth over-dsperso because of ts propert, equ-dsperso. Here we propose a ew dstrbuto for overdspersed cout data, amel the Posso-geeralsed Ldle dstrbuto. Basc propertes of the dstrbuto ad specal cases are also derved. I addto, the ew dstrbuto s appled to some real data sets usg the method of maxmum lkelhood for parameter estmato. The results based o p-value of the dscrete Aderso-Darg test show that the ew dstrbuto ca be used as a alteratve model for cout data aalss. Kewords: cout data, mxed Posso dstrbuto, geeralsed Ldle dstrbuto, over-dsperso 1. Itroducto * Correspodg author. Emal address: fscwb@ku.ac.th Cout data are used to descrbe ma pheomea such as the surace clam umbers, umber of east cells, umber of chromosomes, etc. (Pajer, 6). Cout data aalss ca use a Posso dstrbuto to descrbe the data f ts varace to mea rato, called the dsperso dex, s ut (equ-dsperso) (Johso et al., 5). However, ma practcal cout data sets do ot satf the equ-dsperso assumpto. Therefore, the Posso dstrbuto s flexble to model ma cout data sets (Raghavachar et al., 1997; Karls ad Xekalak, 5). A equalt of varace ad mea s called over-dsperso f the varace exceeds the mea, ad uder-dsperso f the varace s less tha the mea. Ma researchers have looked at the over-dsperso ssue whch ca be addressed b the use of mxed Posso dstrbutos (Raghavachar et al., 1997; Karls ad Xekalak, 5; Pajer, 6). Mxed Posso dstrbutos arse whe the mea of the Posso s a radom varable wth some specfed dstrbuto. The dstrbuto of the Posso rate s the so-called mxg dstrbuto (Evertt ad Had, 1981; Raghavachar et al., 1997). The egatve bomal (NB) dstrbuto, whch s a tradtoal mxed Posso dstrbuto where the mea of the Posso varable s dstrbuted as a gamma radom varable, was derved b Greewood ad Yule (19). It has creasgl become a popular alteratve dstrbuto to the Posso dstrbuto. However, the NB dstrbuto ma ot be approprate for some over-dspersed cout data. Other mxed Posso dstrbutos arse from alteratve mxg dstrbutos. If the mea of the Posso follows a verse Gaussa, resultg a Posso-verse Gaussa (Holla, 1967). The Posso-Ldle (PL) (Sakara, 197) ad geeralsed Posso - Ldle (Mahmoud ad Zakerzadeh, 1) dstrbutos were obtaed where the mxg dstrbutos are the Ldle ad the geeralsed Ldle dstrbutos, respectvel. Recetl, a Posso-weghted expoetal dstrbuto was developed b Zama et al. (14), where a weghted expoetal s the mxg dstrbuto. It has bee foud that the geeral characterstcs of the mxed Posso dstrbuto follow some characterstcs of ts mxg dstrbuto. Depedg o the choce of the mxg dstrbuto, varous mxed Posso dstrbutos have bee

2 646 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , 16 costructed. However, sce ther mathematcal forms are ofte complcated, ol a few of them have bee appled practce. Furthermore, a case, there are aturall stuatos where a good ft s ot obtaable wth exstg developed dstrbutos (Karls ad Xekalak, 5). The purpose of ths paper s to preset a alteratve dstrbuto for over-dspersed cout data, amel the Posso-geeralsed Ldle (PGL) dstrbuto. It s obtaed b mxg the Posso dstrbuto wth a ew geeralsed Ldle (NGL) dstrbuto (Elbatal et al., 13). The probablt dest fucto of the three-parameter NGL dstrbuto, whch geeralsed the Ldle dstrbuto, has ma shapes. Due to ts flexble shape, t ca be used as a alteratve model for fttg postve real-valued data ma areas. For ths paper, we show that the proposed mxed Posso dstrbuto s sutable for modellg real cout data some stuatos. I Secto, the ew dstrbuto, called the PGL dstrbuto, s troduced. Some specal cases of the dstrbuto are also cosdered ths secto. Its basc mathematcal propertes cludg the momet geeratg fucto, probablt geeratg fucto ad momets are derved Secto 3. We also dscuss the method of parameter estmato Secto 4. Fall, applcatos of the PGL to real data are gve Secto 5.. The Posso-Geeralsed Ldle Dstrbuto Let Y be the radom varable that represets the total umber of outcomes of a partcular expermet. The smple model for the probabltes of the possble outcomes of ths expermet s the Posso dstrbuto, wth probablt mass fucto (pmf) exp( ) p( ) ;,1,,, ad. (1)! A mportat propert of the Posso dstrbuto s that the postve real umber equals both the expected value of Y ad ts varace,.e. E( Y ) Var( Y ). I 13, the NGL dstrbuto was troduced (Elbatal et al., 13). It s a three-parameter cotuous dstrbuto used to aalse lfetme data. It ca model ma shapes of hazard rate fucto. The probablt dest fucto (pdf) ca be obtaed b cocept of the fte mxture dstrbuto g( ) pg ( ) (1 p) g ( ) exp( ); 1 ( ) ( ), ad,,, where ~ Gamma(, ), ~ Gamma(, ), p / 1 ( 1), ad the gamma fucto s defed as ( t) t 1 x x x exp( )d. () The PGL dstrbuto s a ew mxed Posso dstrbuto. It s obtaed b mxg the Posso dstrbuto wth the NGL dstrbuto. We provde a geeral defto of ths dstrbuto whch wll subsequetl expose ts pmf. Defto 1: Let Y be a radom varable followg a Posso dstrbuto wth parameter, Y ~ Pos( ). If s dstrbuted as a ew geeralsed Ldle wth parameters, ad, deoted b ~ NGL(,, ), the Y s called a Posso-geeralsed Ldle varable. Proposto 1: Let Y be a radom varable accordg to the PGL probablt fucto, deoted b Y ~ PGL(,, ), the pmf of Y s p( ;,, ) 1 ( ) ( ), (3) 1!( 1) 1 ( ) 1 ( ) for,1,,, ad,,. Proof: Sce Y ~ Pos( ) ad ~ NGL(,, ), the margal pmf of Y ~ PGL(,, ) ca be obtaed b p( ;,, ) p( ) g( )d. (4) B substtutg Eq. (1) ad Eq. () to Eq. (4), we derve the margal pmf of the PGL dstrbuto: p( ;,, ) exp( ) 1 exp( )d! 1 ( ) ( ) 1 1!( 1) ( ) ( ) 1 exp( ( 1) )d 1 exp( ( 1) )d 1 1 ( ) ( ) 1!( 1) ( 1) ( ) ( ) 1 1 ( ) ( ). 1!( 1) 1 ( ) 1 ( ) Moreover, f a radom varable correspodg to, {,1,, } s a sample space, the pmf of Y s the probablt fucto wth the followg propertes:

3 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , I. If a radom varable Y s dstrbuted as the PGL wth the pmf Eq. (3), whe =, we obta 1 ( ) ( ) P( Y ) ( 1) 1 ( ) 1 ( ) for,,, P( Y ). If 1, we have P( Y 1) 1 ( 1) ( 1) ( 1) 1 ( ) 1 ( ) 1 ( 1) ( 1) ( 1) 1 ( ) 1 ( ) 1, 1 1 where,,, the P( Y 1). I the same maer, t s obvousl,,1,, 3,..., the probablt of Y s greater tha or equal to zero. Therefore, P( Y ) sastfes P( Y ), for all Y. II. If a radom varable s dstrbuted as the PGL wth the pmf Eq. (3), the p( ;,, ) 1 1 ( ) ( ) 1!( 1) 1 ( ) 1 ( ) 1 ( ) ( )! ( )( 1)! ( )( 1) ( ) (1 ) ( ) ( )( 1) ( )( 1) ( )( 1) 1 3 ( ) (1 ) ( )( 1) ( )( 1) ( ) ( )( 1) , Hece, P( ) 1. From I ad II, t ca verf that the pmf of Y,, s a probablt fucto. Fgure 1 llustrates pmf plots of the PGL dstrbuto for some selected parameter values. It was foud that the shape of the PGL dstrbuto s charactersed b log-taled behavour ad also that the dstrbuto has the same shape as the NGL dstrbuto wth approprate parameter values. The parameters ad are the shape parameters ad s the rate parameter of the PGL dstrbuto. Furthermore, the PGL s a bmodal dstrbuto whe parameters ad are ver dfferet for approprate values of the parameter as show Fgure..1 Specal cases Ths secto presets some specal cases of the PGL dstrbuto. Corollar 1: For 1,, we obta the PL dstrbuto wth the pmf ( ) f ( ; ). 3 ( 1) The PL dstrbuto s a mxed Posso dstrbuto, whch s a well-kow dscrete dstrbuto. It has bee used prevousl to model cout data (Sakara, 197, Shaker ad Fesshae, 15). Corollar : For r, we obta the NB dstrbuto wth the pmf r 1 r f ( ; r, p) p (1 p). Corollar 3: For 1, we obta the Posso-expoetal or geometrc dstrbuto wth the pmf f ( ; ). 1 ( 1) 3. Some Propertes of the PGL Dstrbuto Ths secto presets some basc mathematcal propertes of the PGL dstrbuto, specfcall the momet geeratg fucto, probablt geeratg fucto ad the kth momet. (5) (6) (7)

4 648 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , 16 Fgure 1. Some umodal pmf plots of the PGL dstrbuto wth specfed parameter values 3.1 Momet geeratg fucto Proposto : Let Y be a radom varable wth the PGL probablt fucto, the momet geeratg fucto (mgf) of Y ~ PGL(,, ) s M Y 1 1 ( t) ; 1 ( exp( t) 1) ( exp( t) 1) exp( t) ( 1). Proof: The mgf of mxed Posso dstrbuto ca be obtaed from M ( t) E(exp( ty )) Y exp( ) exp( t) g( )d,! sce exp( t) exp( ) /! exp( (exp( t) 1)) s the mgf of Posso dstrbuto, the mgf of PGL wll be

5 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , Fgure. Some bmodal pmf plots of the PGL dstrbuto wth specfed parameter values M ( t) exp( (exp( t) 1)) g( )d Y exp( (exp( t) 1)) exp( )d 1 ( ) ( ) ( ) 1 exp( ( exp( t) 1) )d 1 exp( ( exp( t) ( ) 1) )d ( exp( t) 1) ( exp( t) 1) 3. Probablt geeratg fucto Proposto 3: Let Y be a radom varable wth the PGL probablt fucto, the probablt geeratg fucto (pgf) of Y ~ PGL(,, ) s 1 1 H ( s) ; 1 1 s ( 1). 1 ( s 1) ( s 1) Proof: The pgf of mxed Posso dstrbuto ca be obtaed b utlsg the pgf of Posso dstrbuto as follows sce Y H ( s) E( s ) s exp( ) g( )d,! s exp( ) /! exp( (1 s) ), t s the pgf of Posso dstrbuto, the the pgf of PGL wll be H ( s) exp( (1 s) ) g( )d ( ) 1 exp( ( s 1) )d ( ) 1 exp( ( s 1) )d ( s 1) ( s 1). Alteratvel, the pgf of the PGL dstrbuto ca be got b settg s exp( t) the expresso for the mgf.

6 65 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , Momets Proposto 4: Let Y be a radom varable wth the PGL probablt fucto, the kth factoral momet of Y ~ PGL(,, ) s Proof: The kth factoral momet of a mxed Posso dstrbuto ca be foud b sce 1 ( k ) ( k ). k k 1 k 1 ( ) ( ) exp( ) /!, t s the kth momet about org of the Posso the we obta k k k g( )d k k 1 exp( )d 1 ( ) ( ) 1 1 k 1 r+ -1 exp( )d + exp(- )d 1 ( ) ( ) 1 ( k ) ( k ). 1 1 ( ) k k ( ) k k j k ' j The kth momet about the mea s also called the kth cetral momet, E[( Y ) ] ( 1). k k j j j Cosequetl, the frst four cetral momets of Y ~ PGL(,, ) are ( ( ) 1) ( ( 1) ) = ( 1) ( 1) 3 3( ) ( 1) 3( ) ( 1) ( 1) 1 ( 1)( ) 3 ( 1) ( 1)( ) 3 ( 1) 3 1 ( 1) 4 6 ( 1) ( ) 3( ) ( 1) ( 1) ( 1)( ) 3( 1) ( 1)( ) 3 ( 1) 4 ( ) 3 ( 1) 1 ( )( 3)( 1) 6 ( )( 1) 7 ( 1) 3 1,

7 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , ( 1)( )( 3) 6 ( 1)( ) 7 ( 1). 4 3 I partcular, the mea, varace, skewess ad kurtoss of Y ~ PGL(,, ) accordg to ts geeratg fucto, respectvel, are E( Y), ( 1) ( ( ) 1) ( ( 1) ) Va r( Y ) =, ( 1) Skewess( Y ) 3 3/, ad Kurtoss ( Y ) Parameter Estmato A wdel used method of estmatg the parameters of a dstrbuto s b maxmsg the log-lkelhood fucto of parameters,, called maxmum lkelhood estmato (MLE). Let Y, Y,, Y be a depedet ad detcall dstrbuted 1 radom varables whch has the PGL dstrbuto, ad correspod to,,, whch s a radom sample from the PGL 1 populato. Let Θ (,, ) T be the vector of the parameters. The lkelhood fucto of the PGL dstrbuto s 1 ( ) ( ) L( Θ). 1 1!( 1) 1 ( ) 1 ( ) We ca wrte the log-lkelhood fucto as 1 ( Θ) log ( 1) ( ) ( ) ( 1) ( ) ( ) 1 log! ( ) log( 1) log ( ) log ( ) ( ) log( 1), 1 1 ad the frst partal dervatves of the log-lkelhood wth respect to each parameter, called the score fuctos, are 1 ( Θ) ( 1) ( ) ( )( ( ) log ) 1 1 ( 1) ( ) ( ) ( 1) ( ) ( ) ( 1) ( ) ( )( ( ) log( 1)) 1 ( 1) ( ) ( ) ( 1) ( ) ( ) ( ) log( 1), 1 ( Θ) ( 1) ( ) ( )( ( ) log( 1)) 1 1 ( 1) ( ) ( ) ( 1) ( ) ( ) ad ( 1) ( ) ( )( ( ) log ) 1 ( 1) ( ) ( ) ( 1) ( ) ( ) ( ) log( 1), 1 ( Θ) ( ) ( ) ( 1) ( ( 1)( 1)) 1 1 ( 1) ( ) ( ) ( 1) ( ) ( )

8 65 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , ( ) ( ) ( 1) ( ( 1)) ( 1) ( ) ( ) ( 1) ( ) ( ) 1 ( ) ( ) 1, 1 d where ( t) log ( t) s the dgamma fucto. dt The maxmum lkelhood estmators of the PGL dstrbuto ca be acheved b settg the score fuctos equal to zero, gvg the so-called maxmum lkelhood score equatos, ad solvg ths sstem of equatos. I ths case, the score fuctos are olear ad do ot have aaltcal soluto. Istead, maxmum lkelhood estmates ca be obtaed b a umercal method (e.g., Newto-Raphso method, Nelder-Mead method, BFGS method, SANN method, as mplemeted a R fucto mle). 5. Applcatos to Real Data Sets Some real data sets are cosdered to ft wth the proposed dstrbuto (PGL), Posso, NB ad PL dstrbutos. The frst data set s umber of the mstakes copg groups of radom dgts that was used for llustratg the PL dstrbuto b Sakara (197). The secod data set s the umber of mcroucle after exposure at dose 4 (G) of - Irradato. The were couted usg the ctochalas B method ad ftted wth the NB dstrbuto (Pug ad Valero, 6). The thrd data set s a applcato geetcs, the umber of chromatd aberratos (. g cho 1, 4 hours). It had bee ftted prevousl wth the Posso ad the PL dstrbutos, but gve the amout of over-dsperso the data, the PL dstrbuto s a more approprate model (Shaker ad Fesshae, 15). Aother applcato volvg bmodal data s also cosdered ths part. The data set s the umber of Chemopodum album arable lad per quadrat, whch was ft wth the NB dstrbuto (Blss ad Fsher, 1953). We ft ths data set wth the proposed dstrbuto, the NB dstrbuto ad a fve-parameter mxture of two NB dstrbutos (MxtureNB) wth the weghted parameter, where 1, wth pmf ( r ) ( r ) f ( ; r, r, p, p, ) p 1 p (1 ) p 1 p! ( r )! ( r ) r 1 r Descrptve summares of these data are show Table 1. The dex of dsperso s greater tha ut for all data sets, dcatg that all data sets are over-dspersed. I ths work, the SANN method based o bbmle package (Bolker ad Team, 14) of the R programmg laguage (R Core Team, 14), beg a global optmzato, s used to compute the maxmum lkelhood estmates (Nash, 14). Tables, 3, 4 ad 5 preset the results of fttg the dfferet dstrbutos to these real data sets. We use the estmated log-lkelhood (LL) ad Aderso-Darlg (AD) test for dscrete dstrbutos to compare the expected ad observed values of each data set. The AD-test s a emprcal dstrbuto fucto goodess-of-ft test for dscrete data (Choulaka et al., 1994). The ull hpothess s that data follow whatever dstrbuto that s beg tested cludg Posso, NB, PL, MxtureNB, ad PGL wth gve parameter estmates agast the alteratve that data follow some other dstrbutos. The dscrete AD-test ca be obtaed b usg dgof package (Arold ad Emerso, 11) of the R programmg laguage. Table 1. Summar data M Mode Max Mea Dsperso Number of mstakes copg groups Number of mcroucle Number of chromatd aberratos Number of Cheopodum album per quadrat

9 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , Table. The umber of mstakes copg groups of radom dgts The mstakes copg groups Observed values Expected values Posso NB PL PGL Parameter ˆ =.7833 ˆr =.941 ˆ = ˆ =.784 estmates ˆp =.5456 ˆ =.39 ˆ =.398 LL AD-statstc p-value Table 3. The umber of mcroucle The umber of mcroucle Observed values Expected values Posso NB PL PGL Parameter ˆ = 1.13 ˆr = ˆ = ˆ = 9. estmates ˆp =.8517 ˆ =.947 ˆ = LL AD-statstc p-value The ftted dstrbutos for the umber of mstakes copg groups are show Table. It llustrates that the PGL dstrbuto gves the largest LL value. Although, the dffereces betwee LL values are small, but the dstaces from the observed to expected values ad the p-value based o the dscrete AD-test dcate that the ull hpothess caot be rejected at the.5 sgfcat level. It verfes that the mstakes copg groups follows the PGL dstrbuto wth the hghest p-value ad ca model ths data well. The umber of mcroucle are ftted. From the result Table 3, the LL values from the NB ad the PGL dstrbutos are ver smlar. However, the expected values from the PGL dstrbuto are ver close to the observed values, resultg the ull hpothess beg accepted at the.5 level of sgfcace wth p-value Fttg the dstrbutos to the umber of chromatd aberratos data set shows that the PGL dstrbuto gves the largest value of LL (Table 4). Comparg the observed ad expected values demostrates that the PGL dstrbuto aga provdes a good ft to the umber of chromatd aberratos, wth the hghest p-value (.936). I the case of bmodal data, the MxtureNB dstrbuto seems to provde a bt more approprate for the umber of Cheopodum alblum data set. Based o -p-value, t d-

10 654 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , 16 Table 4. The umber of chromatd aberratos (. g cho 1, 4 hours) The umber of chromatd aberratos Observed values Expected values Posso NB PL PGL Parameter ˆ =.5475 ˆr =.65 ˆ =.384 ˆ = estmates ˆp =.5318 ˆ = ˆ = LL AD-statstc p-value Table 5. The umber of Cheopodum album per quadrat The umber of Cheopodum album per quadrat Observed values Expected values NB MxtureNB PGL Parameter estmates ˆr = ˆr = ˆ =.78 ˆp =.3689 ˆp =.9998 ˆ 1 = ˆr = ˆ =.3477 ˆp =.9998 ˆ = 4.3 LL AD-statstc p-value cates that the data follow the mxture of two NB dstrbutos at the.5 level of sgfcace. Due to the expese of extra parameters of the MxtureNB dstrbuto, the PGL dstrbuto wth close LL value ca be chose as a smpler model for fttg ths data set. Fgure 3 shows plot of the observed values ad the expected values related to those show Tables -5 of the PGL dstrbuto. It llustrates that real data are ver close to the PGL dstrbuto. Therefore, the PGL dstrbuto ca be a alteratve model for cout data some stuatos.

11 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , Fgure 3. Results of fttg dstrbutos to real data sets 6. Coclusos I ths work, a ew mxed Posso dstrbuto s troduced. We cosder that the mea of Posso varable s a depedet ad detcall dstrbuted radom varable accordg to a mxg dstrbuto, a ew geeralsed Ldle dstrbuto. The proposed dstrbuto s called the Possogeeralsed Ldle dstrbuto. We have determed varous mathematcal propertes of the Posso-geeralsed Ldle varable, for stace, the probablt mass fucto, momet geeratg fucto, probablt geeratg fucto, the mea, ad the varace. We show that the egatve bomal, Posso-Ldle, ad Posso-expoetal dstrbutos are specal cases of t. The proposed dstrbuto s appled to several real data sets. The results, cludg the p-value based o the dscrete Aderso-Darlg test, dcate that the Possogeeralsed Ldle dstrbuto s a flexble model that ma be a useful alteratve to other dstrbutos for cout data aalss. Ackowledgemets The authors would lke to thak to Departmet of Statstcs, Facult of Scece, ad the graduate school of Kasetsart Uverst. Also we thak to Scece Achevemet Scholarshp of Thalad (SAST) for supportg the frst author. Refereces Arold, T. A. ad Emerso, J. W. 11. Noparametrc goodess-of-t tests for dscrete ull dstrbutos. The R Joural. 3(), Blss, C. I. ad Fsher, R. A Fttg the egatve bomal dstrbuto to bologcal data. Bometrcs. 9(), Bolker, B. ad Team, R. D. C. 14. bbmle: Tools for geeral maxmum lkelhood estmato. R package verso Choulaka, V., Lockhart, R. A., ad Stephes, M. A Cramér-vo mses statstcs for dscrete dstrbutos. Caada Joural of Statstcs. (1), Elbatal, I., Merovc, F., ad Elgarh, M. 13. A ew geeralzed Ldle dstrbuto. Mathematcal Theor ad Modelg. 3, Evertt, B. ad Had, D Fte mxture dstrbutos. Moographs o Statstcs ad Appled Probablt, Chapma ad Hall, U.K. Greewood, M. ad Yule, G. U. 19. A qur to the ature of frequec dstrbutos represetatve of multple happegs wth partcular referece to the occurrece of multple attacks of dsease or of repeated accdets. Joural of the Roal Statstcal Socet. 83(), Holla, M O a Posso-verse Gaussa dstrbuto. Metrka. 11(1),

12 656 W. Wogr & W. Bodhsuwa / Sogklaakar J. Sc. Techol. 38 (6), , 16 Johso, N. L., Kemp, A. W., ad Kotz, S. 5. Uvarate Dscrete Dstrbutos, 3 rd, Wle Seres Probablt ad Statstcs, Joh Wle ad Sos, Ic. Hoboke, New Jerse, U.S.A. Karls, D. ad Xekalak, E. 5. Mxed Posso dstrbutos. Iteratoal Statstcal Revew / Revue Iteratoale de Statstque. 73(1), Mahmoud, E. ad Zakerzadeh, H. 1. Geeralzed Posso - Ldle dstrbuto. Commucatos Statstcs - Theor ad Methods. 39(1), Nash, J. C. 14. O best practce optmzato methods R. Joural of Statstcal Software. 6(), Pajer, H. H. 6. Mxed Posso Dstrbutos. I Ecclopeda of Actuaral Scece, Joh Wle ad Sos, Ltd. Hoboke, New Jerse, U.S.A. Pug, P. ad Valero, J. 6. Cout data dstrbuto: Some characterzatos wth applcatos. Joural of the Amerca Statstcal Assocato. 11(473), R Core Team. 14. R: A Laguage ad Evromet for Statstcal Computg. R Foudato for Statstcal Computg, Vea, Austra. Raghavachar, M., Srvasa, A., ad Sullo, P Posso mxture eld models for tegrated crcuts: A crtcal revew. Mcroelectrocs Relablt. 37(4), Rodríguez-Av, J., Code-Sachéz, A., Saéz-Castllo, A. J., Olmo-Jmeéz, M. J., ad Martíez- Rodríguez, A. M. 9. A geeralzed Warg regresso model for cout data. Computatoal Statstcs ad Data Aalss. 53(1). Sakara, M The dscrete Posso-Ldle dstrbuto. Iteratoal Bometrc Socet. 6(1) Shaker, R. ad Fesshae, H. 15. O Posso-Ldle dstrbuto ad ts applcatos to bologcal sceces. Bometrcs ad Bostatstcs Iteratoal Joural. (4), 1-5. Zama, H., Ismal, N., ad Farough, P. 14. Posso-weghted expoetal uvarate verso ad regresso model wth applcatos. Joural of Mathematcs ad Statstcs. 1(),