A Generalized Poisson-Akash Distribution: Properties and Applications

Transcription

1 Inernaional Journal of Saisics and Applicaions 08, 8(5): DOI: 059/jsaisics A Generalized Poisson-Akash Disribuion: Properies and Applicaions Rama Shanker,*, Kamlesh Kumar Shukla, Tekie Asehun Leonida Deparmen of Saisics, College of Science, Erirea Insiue of Technology, Asmara, Erirea Deparmen of Applied Mahemaics, Universiy of Twene, The Neherlands Absrac A generalized Poisson- Akash disribuion which includes Poisson-Akash disribuion has been proposed Is facorial momens, raw momens and cenral momens have been derived and sudied Some saisical properies including generaing funcions, hazard rae funcion and unimodaliy have been discussed Mehod of momens and he mehod of maimum likelihood have been discussed for esimaing parameers of he disribuion Applicaions of he proposed disribuion have been eplained hrough wo coun daases and compared wih oher discree disribuions Keywords Generalized Akash disribuion, Poisson- Akash disribuion, Compounding, Momens, Skewness, Kurosis, Maimum likelihood esimaion, Applicaions Inroducion The saisical analysis and modeling of coun daa are crucial in almos all fields of knowledge including biological science, insurance, medical science, finance, sociology, psychology, are some among ohers Coun daa are generaed by many phenomena such as he number of insurance claimans in insurance, number of yeas cells in biological science, number of chromosomes in geneics, ec I has been observed ha, in general, coun daa follows under-dispersion (variance < mean), equi-dispersion (variance = mean) or over-dispersion (variance > mean) The over-dispersion of coun daa have been addressed using mied Poisson disribuions by differen researchers including Raghavachari e al (997), Karlis and Xekalaki (005), Panjeer (006), some among ohers Mied Poisson disribuions arise when he parameer of he Poisson disribuion is a random variable having some specified disribuions The disribuion of he parameer of he Poisson disribuion is known as miing disribuion I has been observed ha he general characerisics of he mied Poisson disribuion follow some characerisics of is miing disribuions Various mied Poisson disribuions have been derived in saisics by selecing differen miing disribuion The classical negaive binomial disribuion (NBD) derived by Greenwood and Yule (90) is he mied Poisson disribuion where he parameer of he Poisson random variable is disribued as a gamma random variable The NBD has been used o model over-dispersed coun daa However, he NBD may no be suiable for some over-dispersed coun daa due o is heoreical or applied poin of view Oher mied Poisson disribuions arise from alernaive miing disribuions For eample, he Poisson-Lindley disribuion, inroduced by Sankaran (970), is a Poisson miure of Lindley (958) disribuion The Poisson-Akash disribuion, inroduced by Shanker (07), is a Poisson miure of Akash disribuion proposed by Shanker (05 I has been observed by Karlis and Xekalaki (005) ha here are naurally arising siuaions where a good fi is no obainable wih a paricular mied Poisson disribuion in case of over-dispersed coun daa This shows ha here is a need for new mied Poisson disribuion which gives a beer fi as compared wih he eising mied Poisson disribuions Shanker (07) inroduced he discree Poisson- Akash disribuion (PAD) o model coun daa and defined by is probabiliy mass funcion (pmf) * Corresponding auhor: shankerrama009@gmailcom (Rama Shanker) Published online a hp://journalsapuborg/saisics Copyrigh 08 The Auhor(s) Published by Scienific & Academic Publishing This work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY) hp://creaivecommonsorg/licenses/by/40/

2 50 Rama Shanker e al: A Generalized Poisson-Akash Disribuion: Properies and Applicaions P ; ; 0,,,, 0 Momens and momens based measures, saisical properies; esimaion of parameer using boh he mehod of momens and he mehod of maimum likelihood and applicaions of PAD has been discussed by Shanker (07) The disribuion arises from he Poisson disribuion when is parameer follows Akash disribuion inroduced by Shanker (05) and defined by is probabiliy densiy funcion (pdf) The pdf () is a conve combinaion of eponenial f, e ; 0, 0 and gamma () (), disribuions Shanker (05) discussed saisical properies including momens based coefficiens, hazard rae funcion, mean residual life funcion, mean deviaions, sochasic ordering, Renyi enropy measure, order saisics, Bonferroni and Lorenz curves, sress- srengh reliabiliy, along wih esimaion of parameer and applicaions o model lifeime daa from biomedical science and engineering The firs four momens abou origin and he variance of PAD () obained by Shanker (07) are given by Recenly Shanker e al (08) proposed a generalized Akash disribuion (GAD) having parameers and and defined by is pdf f ;, e ; 0, 0, 0 Is srucural properies including momens, hazard rae funcion, mean residual life funcion, mean deviaions, sochasic ordering, Renyi enropy measure, order saisics, Bonferroni and Lorenz curves, sress- srengh reliabiliy, esimaion of parameers and applicaions for modeling survival ime daa has been discussed by Shanker e al (08) I can be easily shown ha a, GAD () reduces o Akash disribuion () The main purpose of his paper is o inroduce a generalized Poisson- Akash disribuion, a Poisson miure of generalized Akash disribuion proposed by Shanker e al (08) Is momens based measures including coefficiens of variaion, skewness, kurosis and inde of dispersion have been derived and heir naure and behavior has been discussed graphically Is saisical properies including generaing funcions, hazard rae funcion and unimodaliy have been discussed The esimaion of parameers has been discussed using mehod of momens and he mehod of maimum likelihood Applicaions and goodness of fi of he disribuion have also been discussed hrough wo eamples of observed real coun daases and he fi has been found quie saisfacory over oher discree disribuions A Generalized Poisson- Akash Disribuion Assuming ha he parameer of he Poisson disribuion follows GAD (), he Poisson miure of GAD can be obained as ()

3 Inernaional Journal of Saisics and Applicaions 08, 8(5): e P ;, e d 0 () e d e d 0 0 ; 0,,,, 0, 0 We would call his pmf a generalized Poisson - Akash disribuion (GPAD) I can be easily verified ha PAD () is a paricular case of GPAD for The naure and behavior of GPAD for varying values of he parameers and have been eplained graphically in figure () Figure Probabiliy mass funcion plo of GPAD for varying values of parameers and

4 5 Rama Shanker e al: A Generalized Poisson-Akash Disribuion: Properies and Applicaions Momens Facorial Momens Using (), he r h facorial momen abou origin of he GPAD () can be obained as r E E X r r, where X X X X X r r e e d 0! Taking r y, we ge r r e e d 0 r r y r e r e d 0 y 0 y! r e d 0 r r r! r ; r,,, Taking r,,, and 4 in (), he firs four facorial momens abou origin of GPAD () can be obained as Momens abou Origin (Raw momens) () The firs four momens abou origin, using he relaionship beween facorial momens abou origin and he momens abou origin, of GPAD () can be obained as

5 Inernaional Journal of Saisics and Applicaions 08, 8(5): Momens abou he Mean (Cenral momens) r r r Using he relaionship r EY k k0 k abou origin, he momens abou he mean of he GPAD () can be obained as 8 4 rk beween momens abou he mean and he momens Coefficien of Variaion, Skewness, Kurosis and Inde of Dispersion The coefficien of variaion CV, coefficien of Skewness dispersion of he GPAD () are hus obained as CV 8 4 6, coefficien of Kurosis Now from he inde of dispersion i is obvious ha if and 0, hen and inde of (over dispersion) and hence GPAD is a suiable model for over dispersed daa Naure and behavior of coefficien of variaion, coefficien of skewness,

6 54 Rama Shanker e al: A Generalized Poisson-Akash Disribuion: Properies and Applicaions coefficien of kurosis and inde of dispersion of GPAD for varying values of parameers graphically in figure and have been shown Figure Naure and behavior of coefficien of variaion, coefficien of skewness, coefficien of kurosis and inde of dispersion of GPAD for varying values of parameers and 5 Saisical Properies 5 Generaing Funcions The probabiliy generaing funcion of GPAD can be obained as PX The momen generaing funcion of GPAD is hus given by M X e 4 e e e e

7 Inernaional Journal of Saisics and Applicaions 08, 8(5): Increasing Hazard Rae and Unimodaliy We have P ;, P ;, I can be easily verified ha his is a decreasing funcion in, and hence P ;, is log-concave Now using he resuls of relaionship beween log-concaviy, unimodaliy and increasing hazard rae (IHR) of discree disribuions available in Grandell (997), i can concluded ha GPAD () has an increasing hazard rae and is unimodal 6 Parameer Esimaion 6 Mehod of Momens Since GPAD has wo parameers o be esimaed, aking he firs wo momens abou origin, we ge Assuming b, we ge This gives a quadraic equaion in b as 6 b b b 6 k kb kb k k(say) Replacing he firs populaion momen abou origin and he second populaion momen abou origin wih heir respecive sample momens, an esimae of k can be obained and subsiuing he value of k in he above equaion, an esimae of b can be obained Again, replacing he populaion mean wih he corresponding sample mean and aking b in 6, we ge b 6 b b 6 Thus MOME of can be epressed as b b 6 Maimum Likelihood Esimaion Le, which gives MOME of as,,, n be a random sample of size n from he GPAD () and le sample corresponding o X (,,,, k) such ha non-zero frequency The log- likelihood funcion of GPAD () can be given by The maimum likelihood esimaes likelihood equaions k f b 6 b f be he observed frequency in he n, where k is he larges observed value having k log L n log log f log k f log ˆ, ˆ of parameers, of GPAD () is he soluions of he following log-

8 56 Rama Shanker e al: A Generalized Poisson-Akash Disribuion: Properies and Applicaions k log L n n n f 0 f 0 k log L n where is he sample mean These wo log- likelihood equaions do no seem o be solved direcly because hey are no in closed forms These wo log-likelihood equaions can be solved ieraively using R-sofware ill sufficienly close values of ˆ and ˆ are obained The iniial values of parameers are aken as ˆ 05 and ˆ 05 7 Applicaions The GPAD has been fied o wo coun daases from biological sciences o es is goodness of fi over Poisson disribuion (PD), Poisson-Lindley disribuion (PLD) and Poisson-Akash disribuion (PAD) The maimum likelihood esimae (MLE) has been used o fi he GPAD The firs daase is he number of Suden s hisoric daa on Haemocyomeer couns of yeas cells available in Gosse (908) and he second daa se is he number of European corn- borer of Mc Guire e al (957) The fied plos of disribuions for daases in ables and are presened in figure I is clear from he goodness of fi of GPAD and from he fied plos of disribuions ha GPAD gives much closer fi han PD, PLD, and PAD and hence i can be considered as an imporan disribuion in ecology Table Observed and epeced number of Haemocyomeer yeas cell couns per square observed by Gosse (908) Number of yeas cells per square Observed frequency Epeced frequency PD PLD PAD GPAD Toal ML Esimaes ˆ 0685 ˆ 950 ˆ 60 ˆ ˆ df p-value log L AIC Number of corn- borer per plan Table Observed and epeced number of European corn- borer of Mc Guire e al (957) Observed frequency Epeced frequency PD PLD PAD GPAD Toal ML Esimaes ˆ 0648 ˆ 04 ˆ ˆ 787 ˆ 8887

9 Inernaional Journal of Saisics and Applicaions 08, 8(5): df p-value log L AIC The fied plo of disribuions for daases in able and are shown in figure I is obvious from he goodness of fi in ables and and he fied plo of disribuions in figure ha GPAD gives beer fi Figure Fied plos of disribuions for daases and 8 Concluding Remarks This paper proposes a generalized Poisson- Akash disribuion which includes Poisson-Akash disribuion as a paricular case Is momens and momens based measures have been derived and sudied Some saisical properies have been discussed Mehod of momens and he mehod of maimum likelihood have been discussed for esimaing parameers of he disribuion Finally, applicaions of he proposed disribuion have been eplained hrough wo coun daases from biological sciences and he goodness of fi has been found quie saisfacory over PD, PLD and PAD ACKNOWLEDGEMENTS Auhors are graeful o he edior in chief and he anonymous reviewer for heir consrucive commens which enhanced he qualiy of he paper REFERENCES [] Grandell, J (997): Mied Poisson Processes, Chapman & Hall, London [] Greenwood, M and Yule, GU (90): An inquiry ino he naure of frequency disribuions represenaive of muliple happenings wih paricular reference o he muliple aacks of disease or of repeaed accidens, Journal of he Royal Saisical Sociey, 8(), 5- [] Gosse, WS (908): The probable error of a mean, Biomerika, 6, -5 [4] Karlis, D and Xekalaki, E (005): Mied Poisson disribuions, Inernaional Saisical review, 7(), 5-58 [5] Lindley, DV (958): Fiducial disribuions and Bayes heorem, Journal of he Royal Saisical Sociey, 0 (), 0-07 [6] Mc Guire, JU, Brindley, TA and Bancrof, TA (957): The disribuion of European corn-borer larvae pyrausa in field corn, Biomerics,, [7] Panjeer, H H (006): Mied Poisson disribuions In Encyclopedia of Acuarial Science, John Wiley and Sons Ld, Hoboken, New Jersey, USA

10 58 Rama Shanker e al: A Generalized Poisson-Akash Disribuion: Properies and Applicaions [8] Raghavachari, M, Srinivasam, A and Sullo, P (997): Poisson miure yield models for inegraed circuis A criical review, Microelecronics Reliabiliy, 7 (4), [9] Sankaran, M (970): The discree Poisson-Lindley disribuion, Biomerics, 6, [0] Shanker, R (05): Akash disribuion and Is Applicaions, Inernaional Journal of Probabiliy and Saisics, 4(), [] Shanker, R (07): The Discree Poisson-Akash Disribuion, Inernaional Journal of Probabiliy and Saisics, 6(), -0 [] Shanker, R Shukla, KK, Shanker, R, Singh, AP (08): A Generalized Akash Disribuion, Biomerics & Biosaisics Inernaional Journal, 7(), -0