ZERO - ONE INFLATED POISSON SUSHILA DISTRIBUTION AND ITS APPLICATION

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1 ZERO - ONE INFLATED POISSON SUSHILA DISTRIBUTION AND ITS APPLICATION CHOOKAIT PUDPROMMARAT Depatmet of Sciece, Faculty of Sciece ad Techology, Sua Suadha Rajabhat Uivesity, Bagkok, Thailad chookait.pu@ssu.ac.th Abstact - I this pape, we peset a zeo oe iflated Poisso Sushila (ZOIPS distibutio which is discete distibutio. Seveal statistical popeties of ZOIPS distibutio ae exploed, such as the pobability mass fuctio (pmf, momet about the oigi, mea ad vaiace. Maximum likelihood estimatio of the paametes is ivestigated. A applicatio of the distibutio to a eal data set is peseted fially ad compaed with the fit attaied by some othe well-kow distibutios fo cout data. Idex Tems - Poisso - Sushila distibutio; Zeo-oe iflated model; Discete distibutio. I. INTRODUCTION The Poisso distibutio is a well -kow distibutio to fit cout data i pactice. Howeve theoetical pedictio may ot match empiical obsevatios fo momet of highe ode due to the oly oe paamete, which does ot allow the vaiace to be adjusted idepedetly of mea []. Fo this poblem, the mixed distibutio has become iceasigly popula as a moe flexible alteative to the Poisso distibutio. Especially it is doubtful whethe the stict equiemets, paticulaly idepedece, fo the Poisso distibutio will be satisfied []. Mixed distibutios defie oe of the most impotat ways to obtai ew pobability distibutios i applied pobability ad opeatioal eseach []. Oe mixed distibutio has bee poposed i applicatio to cout data, especially mixed Poisso. Examples of mixed Poisso ae Poisso-uifom [4], Poisso - logomal [5], Poisso-ivese gamma [6], Poisso-Paeto [6], Poisso-Lowmax [7], Poisso- gamma[], etc. These have bee poposed i fittig cout data ude the ovedispesio. Sushila distibutio was itoduced by Shake, R et al. [8] which is a mixed distibutio betwee expoetial distibutio ad gamma distibutio. Sushila distibutio is a two-paamete cotiuous distibutio. The popeties of Sushila distibutio such as momets, failue ate fuctio, mea esidual life fuctio, stochastic odeigs, estimatio of paametes by the method of maximum likelihood ad the method of momets. The Poiso - Sushila distibutio was itoduced by Saatoo [9] which is a two paamete discete distibutio. Vaious popeties such as momets, mea, vaiace, skewess ad kutosis ad estimatig paamete by usig maximum likelihood estimatio have bee studied ad show that the Poisso -Sushila distibutio is moe flexible tha Poisso distibutio i eal data. A extesio to the zeo-iflated powe seies distibutios o zeo - oe iflated models was itoduced by Alshkaki []. Poisso distibutio is oe of powe seies distibutio. A extesio to the zeo-iflated powe seies distibutios is discete distibutios with umbe of fequecies with zeo ad oe ae iflated. Alshkaki [] studied its stuctue popeties; its mea, vaiace, pobability geeatig fuctio. Zeo - oe iflated Poisso distibutio was itoduced by Alshkaki []. The zeo - oe iflated Poisso distibutio is a extesio to the case of zeo iflated Poisso distibutio. Its stuctual popeties wee studied such as mea, vaiace ad estimatio of its paametes usig the methods of momets ad maximum likelihood estimatos. I this study, we itoduce a ew discete distibutio, which is the zeo oe iflated Poisso - Sushila (ZOIPS distibutio. Additioally, we peset some popeties of the ZOIPS distibutio ae the momet about oigi, mea ad vaiace. The paametes of ZOIPS distibutio ae estimated by maximum likelihood method, ad we also peset fittig distibutio betwee the zeo iflated Poisso, zeo oe iflated Poisso ad ZOIPS distibutios based o eal data set. II. ZERO-ONE INFLATED POISSON - SUSHILA DISTRIBUTION This sectio, we peset zeo oe iflated Poisso - Sushila (ZOIPS distibutio. Defiitio. Let X ZOIPS (,, p, p be a adom vaiable of ZOIPS distibutio with paamete,, p ad p. Theoem. Let X ZOIPS (,, p, p The pobability mass fuctio (pmf of X is give by Poceedigs of Reseachfoa 5 th Iteatioal Cofeece, Hambug, Gemay, d - d Febuay 8 88

2 ( p ( p p, x ( ( ( f ( x p ( p p, x ( ( x ( x ( p p, x,,... x ( ( Zeo - Oe Iflated Poisso Sushila Distibutio ad Its Applicatio whee,, p ad p. Poof. If X is a adom vaiable of Poisso - Sushila, the the pmf of X ca be obtaied by x x g( x x ( ( x,,,,,. The zeo oe iflated model is a exta popotio added to popotio of zeo, the the pmf of zeo-oe iflated model is defied by p ( p p g( x, x f ( x p ( p p g( x, x ( p p g( x, x,,... whee p ad p. The, the pmf of the ZOIPS is obtaied by subsitutig the pobability desity fucito of Poisso - Sushila adom vaiable ito zeo oe iflated model. Fially, it ca be witte as ( p ( p p, x ( ( ( f ( x p ( p p, x ( ( x ( x ( p p, x,,... x ( ( Next, We display some of pobability mass fuctio of ZOIPS i Figue -. Figue. The Pmf of a ZOIPS Radom Vaiable of Some Values of Paametes:, p., p.. Next, we will epeset the Poisso-Sushila, zeo iflaed Poisso-Sushila ad Oe iflated Poisso - Sushila distibutios as a special case of the ZOIPS distibutio i Coollay -. Coollay. If p ad p the the ZOIPS distibutio educes to the Poisso-Sushila distibutio with pmf give by x ( x f ( x, x,,,,,. x ( ( Coollay. If p the the ZOIPS distibutio educes to the oe iflated Poisso-Sushila distibutio with pmf give by ( p ( p, x ( ( f ( x x ( x ( p, x,,,... x ( ( whee, ad p. Coollay. If p the the ZOIPS distibutio educes to the zeo iflated Poisso-Sushila distibutio with pmf give by ( p ( p, x ( ( f ( x x ( x ( p, x,,,... x ( ( whee, ad p. III. SOME PROPERTIES OF ZERO-ONE INFLATED POISSON - SUSHILA DISTRIBUTION Figue. The pmf of a ZOIPS adom vaiable of some values of paametes:, p., p.. I this pat, we itoduce some popeties of the ZOIPS distibutio. We begi with momet about the oigi of this distibutio i Theoem. Poceedigs of Reseachfoa 5 th Iteatioal Cofeece, Hambug, Gemay, d - d Febuay 8 89

3 Theoem. If X ZOIPS (,, p, p, the the th Zeo - Oe Iflated Poisso Sushila Distibutio ad Its Applicatio momet about oigi of X is give by!( E( X p ( p p ( whee,,,,, p ad p. Poof. Fom the th momet about oigi of ZOIPS ca be obtaied by E( X x f ( x x f ( x x x x ( x E( X p x ( p p x ( ( x p ( p p We have that x ( x ( (. x x x x ( x is the th x ( ( momet about oigi of Poisso - Sushila distibutio. The,!( E( X p ( p p (. Fom the th momet about oigi of ZOIPS distibutio, it is staightfowad to deduce the fist - fouth momet about oigi, mea, ad vaiace espectively ae fist - fouth momet about oigi ( E( X p ( p p ( ( 6 E( X p ( p p ( ( 6 4 E( X p ( p p ( 4 ( 5 E( X p ( p p ( Mea momet ( E( X p ( p p ( Vaice ( ( Va ( X p ( p p p ( p p ( ( 4 Mea ad vaiace of pobability mass fuctio of ZOIPS distibutio is show i Figue. 4 (b Figue. Mea (a ad Vaiace (b of a ZOIPS Radom Vaiable of Some Values of Paametes. (a IV. PARAMETER ESTIMATION I this sectio, the estimatio of paametes fo the ZOIPS via the maximum likelihood pocedue is povided. The likelihood fuctio of the ZOIPS (,, p, p is give by ( L(,, p, p p ( p p i ( p ( p p x ( x ( p p x ( ( ai bi ai bi, xi, xi whee ai ad bi, xi, xi with the coespodig log- likelihood fuctio: ( log L(,, p, p ai log p ( p p i ( bi log p ( p p i a b log( p p i i i i i a b log a b log( i i i i a b ( x log( a b x log i i i i i i i i a b log( x. i i i i Poceedigs of Reseachfoa 5 th Iteatioal Cofeece, Hambug, Gemay, d - d Febuay 8 9

4 The optimal values of the paametes ca be obtaied by the fist patial deivatives of log- likelihood fuctio with espect to,, p ad p. The, it gives ise to followig equatios: ( ( ( log L ( p ( p p Zeo - Oe Iflated Poisso Sushila Distibutio ad Its Applicatio ( ( ( ( p ( p p ai bi xi ( w i ai bi x i i hee a, b, ( ( ad i i i i ( ( ( log L ( p ( p p ( ( ( p ( p p ai bi xi i i ai bi x i i a b x ( log L ( ( p ( p ( p p i i i ( ( ( ( p p p ( p p ( log L ( ( p ( p ( p p ( ( ( ( p p p ( p p These ca be solved umeically by usig Newto-Raphso method. I this pape, we obtai the MLE estimates of ˆ, ˆ, pˆ ad p ˆ. by usig fuctio lm i stats package of R pogam. V. APPLICATION The ZOIPS distibutio is applied o cout data. We use a eal data set which is the umbe of stillbiths i 4 littes of New Zealad white abbits, oigially used by Moga et al []. The eal data set is fitted by the zeo iflated Poisso (ZIP, zeo - oe iflated Poisso (ZOIP ad ZOIPS distibutios. The maximum likelihood method povides paamete estimatios. The pefomaces of the model fittigs by usig chi-squaed statistic ad Akaike ifomatio citeio (AIC i Table. The esults show that the ZOIPS distibutio povides a bette fit tha the ZIP ad ZOIP distibutios. CONCLUSION I this study, we itoduce the ZOIPS distibutio which is a exta popotio of zeo of the adom vaiable Poisso - Sushila. I paticula, the closed fom, fist - fouth momet about oigi, mea, ad vaiace of the ZOIPS distibutio ae deived. I additio, the paamete estimatios ae show via the maximum likelihood method. Fo applicatio to eal data set, it shows that this distibutio ca povide a bette fit tha othe paet distibutios. We hope that the ZOIPS distibutio may be a alteative model of cout data aalysis with may zeo ad oe. Table. Estimated paametes of ZIP, ZOIP ad ZOIPS fo umbe of stillbiths of New Zealad white abbits data. ACKNOWLEDGEMENTS This eseach was successful by fiacial suppot by Sua Suadha Rajabhat Uivesity ad may helps fom my fieds, colleague, ad studets fom Depatmet of Sciece, Faculty of Sciece ad Poceedigs of Reseachfoa 5 th Iteatioal Cofeece, Hambug, Gemay, d - d Febuay 8 9

5 Techology. The autho eally appeciates ad may thaks fo all. REFERENCES [] Wag, Z., Oe mixed egative biomial distibutio with applicatio, Joual of Statistical Plaig ad Ifeece, vol. 4, o., pp. 5 6,. [] Johso, L.N., Kemp, W.A ad Kotz, S., Discete Uivaiate Distibutios d ed., New Yok, Joh Wiley, 5. [] Gómez-Déiz, E., Saabia, J.M. & Calde ı-ojeda, E., Uivaiate ad multivaiate vesios of the egative biomial ivese Gaussia distibutios with applicatios, Mathematics ad Ecoomics, vol. 4, pp. 9 49, 8. [4] Bhattachaya, S.K., Cofluet Hypegeometic Distibutios of Discete ad Cotiuous type with Applicatio to Accidet Poeess, Bulleti of the Calcutta Statistical Associatio, vol. 5, pp., 966. [5] Bulme, M.G., O Fittig the Poisso Logomal Distibutio to Species Abudace Data, Biometics, vol., pp., 974. Zeo - Oe Iflated Poisso Sushila Distibutio ad Its Applicatio [6] Willmot, G.E., O Recusive Evaluatio of Mixed Poisso Pobabilities ad Related Quatities, Scadiavia Actuaial, vol. 8, pp. 4, 99. [7] Al Awadhi ad Ghitay, Statistical Popeties of Poisso Lomax Distibutio ad its Applicatio to Repeated Accidets Data, Applied Statistical Sciece, vol., pp. 65 7,. [8] Shake, R., Shambhu S., Uma S. ad Ravi S., Sushila distibutio ad Its applicatio to waitig times data, Iteatioal Joual of Busiess Maagemet, vol., o., pp. -, Dec.. [9] Saatoo, A., Poisso Sushila distibutio ad its applicatio, B.Sc.(Applied Statistics Reseach Poject, Sua Suadha Rajabhat Uivesity, Thailad, 7. [] Alshkaki, R. S. A., A Extesio to the Zeo-Iflated Geealized Powe Seies Distibutios, Iteatioal Joual of Mathematics ad Statistics Ivetio, vol. 4, o. 9, pp , 6. [] Alshkaki, R. S. A., O the Zeo-Oe Iflated Poisso Distibutio, Iteatioal Joual of Statistical Distibutios ad Applicatios, vol., o. 4, pp. 4-48, 6. [] Moga, B. J. T., Palme, K. J., ad Ridout, M. S., Negative Scoe Test Statistics, The Ameica Statisticia, vol.6, pp , 7. Poceedigs of Reseachfoa 5 th Iteatioal Cofeece, Hambug, Gemay, d - d Febuay 8 9