The Alpha-Logarithmic Series Distribution of Order k and Some of Its Applications

Transcription

1 oual of Saisical Theo ad Applicaios Vol. 5 No. 3 Sepembe The Alpha-Logaihmic Seies Disibuio of Ode ad Some of Is Applicaios C. Saheesh Kuma Depame of Saisics Uivesi of Keala Tivadum Idia dcsaheeshuma@gmail.com A. Riaz Depame of Saisics Uivesi of Keala Tivadum Idia iazsaoou@gmail.com Received Novembe 4 Acceped 4 Apil 6 Absac ee we develop a ode vesio of he alpha-logaihmic seies disibuio of Kuma ad Riaz Souh Afica Sais.. 4 hough is pobabili geeaig fucio ad deive is pobabili mass fucio mea ad vaiace. The paamees of he model ae esimaed b he mehod of maximum lielihood ad he disibuio has bee fied o ceai eal life daa ses. Ceai es pocedues ae cosideed fo esig he sigificace of he addiioal paamees of he disibuio. I addiio a simulaio sud is coduced fo assessig he pefomace of he lielihood esimaos of each of he paamees of he model. Kewods: Cou daa models; geealized lielihood aio es; logaihmic seies disibuio; maximum lielihood esimaio; Maov chai Moe Calo simulaio; pobabili geeaig fucio; Rao s efficie scoe es. Mahemaics Subec Classificaio: 6E 53C35. Ioducio The logaihmic seies disibuio LSD was ioduced as he limi of a zeo-ucaed egaive biomial disibuio b Fishe e al. [4] o ivesigae he disibuio of bueflies i he Malaa Peisula. The LSD has bee used exesivel b Williams [8 9] fo fiig ecological daa o species of abudace wih log ails. A applicaio of he LSD o a iveo cool poblem i seel idus appeas i Williamso ad Beheo [3]. Chafield e al. [] uilized he LSD o epese he disibuio of umbe of iems of a poduc puchased b a bue i a specified ime peiod. The LSD has bee foud applicaio i seveal pacical siuaios such as he sudies o samplig of quadas fo pla species he disibuio of aimal species populaio gowh biolog ecoomics iveo models ad maie scieces. Fo deails see oso e al. [6]. Due o pacical suiabili of he LSD i case of daa wih log ails seveal geealized foms of i have bee poposed i he lieaue. Fo example see ai ad Gupa [5] Kempo [7] Tipahi ad Gupa [6 7] Og [] Khag ad Og [8] ad Kuma ad Riaz [4 5]. 73

2 C. Saheesh Kuma ad A. Riaz A impoa dawbac of he LSD i ceai pacical siuaios is ha i excludes he zeo obsevaio fom is suppo. To miigae his dawbac Khai [9] has cosideed a disibuio amel he logaihmic-wihzeos disibuio LWZD hough he followig pobabili mass fucio pmf i which < ω < ω f x x ω x l x x 3... Some aspecs of he LWZD ae also deailed i ohso e al. 6 pp Kuma ad Riaz [6] cosideed a modified fom of he LSD wih a o-egaive suppo amel he alpha- logaihmic seies disibuio ALSD hough he followig pmf i which A [ l ] > ad such ha <. Al g x x x A x x x 3... The pobabili geeaig fucio pgf of he ALSD. wih pmf is he followig... G Al.3 Recel hee has bee eewed iees i he sud of discee disibuios of ode i he lieaue. Fo example see Kuma [ ] Kuma ad Shibu [8] o Kuma ad Nai [ 3]. Fo a deailed accou of ode disibuios ad hei applicaios see secio.7 of ohso e al. [6]. Philippou [] ioduced ad sudied a egaive biomial disibuio of ode which is also ow as he pe I waiig ime disibuio of ode. As a limiig fom of he lef-ucaed vesio of his egaive biomial disibuio of ode a logaihmic disibuio of ode is poposed i he lieaue as give i page 46 of ohso e al. [6]. Paaeos ad Xealai [] obaied aohe logaihmic disibuio of ode as a limiig fom of he gamma-mixed Poisso disibuio of ode. Kuma ad Riaz [7] ioduced ad sudied a ode vesio of he logaihmic seies disibuio which he obaied as he limiig case of he zeo-ucaed cluse egaive biomial disibuio. All hese ode vesios of logaihmic disibuio do o have oegaive suppo. Though he pese pape we coside a ode vesio of he ALSD which possess a o-egaive suppo ad amed i as he alpha-logaihmic seies disibuio of ode o i sho he ALSD. I secio we pese he defiiio of he ALSD ad deive is pmf mea ad vaiace. I secio 3 we esimae he paamees of he ALSD b he mehod of maximum lielihood ad i secio 4 we illusae is usefuless wih he help of ceai eal life daa ses. I secio 5 we coside he geealized lielihood aio es ad Rao s efficie scoe es fo esig he sigificace of he addiioal paamees of he model ad i secio 6 we caied ou a simulaio sud fo compaig he pefomace of he esimaos obaied b he mehod of maximum lielihood. We eed he followig seies epeseaio i he sequel. Fo a eal valued fucio Q s i which [a] deoe he iege pa of a ad Q s Q s.4 s s s u b b u.5 fo a b > ad u <. 74

3 . Defiiio ad Popeies The alpha-logaihmic seies disibuio of ode I his secio fis we pese he defiiio of he ALSD ad discuss some of is impoa popeies. Defiiio. A o-egaive iege valued adom vaiable Y is said o follow he alpha- logaihmic seies disibuio of ode o i sho he ALSD if is pgf is of he followig fom i which Λ [ l ] > ad such ha <. Λ l. Cleal whe he pgf. educes o he pgf of he ALSD of Kuma ad Riaz [6] ad whe he pgf. educes o he pgf of he of he logaihmic seies disibuio of ode sudied b Kuma ad Riaz [7]. The pgf of ALSD give i. ca also be wie as whee F ; ;. F ; ; F a b ; c ; z a b c is he Gauss hpegeomeic fucio i which d d d d fo ad d fo a d R. Fo deails egadig Gauss hpegeomeic fucio see Mahai ad aubold [9] o Slae [5]. Nex we deive he pmf of he ALSD hough he followig heoem. Theoem. The pmf q P Y of he ALSD wih pgf. is he followig. i which Λ l q Λ q z.3 deoe he summaio ove all -uples of o-egaive ieges i he se { : } ad. Poof. Fom. we have he followig..4 q.5 Λ l..6 75

4 C. Saheesh Kuma ad A. Riaz Expad he logaihmic fucio i.6 o ge Λ Λ Λ.7 b biomial heoem. Now o spliig he ie summaio of.7 we obai. Λ Λ.8 B applig.4 i he fis em ad sice he secod em is he logaihmic expasio of ] l [ we have he followig fom.8.. l Λ Λ.9 Now b applig he muliomial expasio i.9 we have l δ Λ Λ. i which δ deoe he summaio ove all uples... of o-egaive ieges i he se } : {. O eaagig he ems we have l... δ Λ Λ. which implies he followig i he ligh of he elaio v p v v p ad he defiiio of he Gauss hpegeomeic fucio. l... δ Λ Λ. l... δ Λ Λ.3 76

5 The alpha-logaihmic seies disibuio of ode i he ligh of.5. Now o equaig he coefficies of o he igh had side expessios of.5 ad.3 we ge.4. Nex we obai he mea ad vaiace of he ALSD hough followig heoem. Theoem. The mea ad vaiace of he ALSD ae he followig i which δ λ. ad ad Mea Λδλ Vaiace Λδ[ λ Λδ λ ]. Poof follows fom he fac ha ad whee fo... Mea Vaiace [ ] d. d 3. Maximum Lielihood Esimaio I his secio we discuss he esimaio of he paamees of he ALSD b he mehod of maximum lielihood esimaio. Le a be he obseved fequec of eves ad le z be he highes value of obseved. The he lielihood fucio L of he sample is he followig i which q is he pmf of he ALSD as give i.4. z L Π [ q ] Taig logaihm o boh sides of 3. we have whee Λ is as give i.4 ad a a l L a l[ q] a l[ q ] z z z a [ q] Π[ q ] 3. a l[ q] a [l Λ lφ ; q q q a] 3. φ ;. 77

6 C. Saheesh Kuma ad A. Riaz whee is as defied i.4. Le ˆ deoe he maximum lielihood esimao of he paamee fo ad ˆ deoe he maximum lielihood esimao of he paamee of he ALSD. O diffeeiaig 3. paiall wih espec o he paamees fo... ad ad equaig o zeo we ge he followig ssem of lielihood equaios. z Λδ [ φ ; ] 3.3 ad z [ Λδ ] [ l φ ; ] Now o solvig he lielihood equaios give i 3.3 ad 3.4 b usig mahemaical sofwaes such as MATLAB MATCAD MATEMATICA ec. we ca obai he maximum lielihood esimaos of he paamees of he ALSD. Whe he lielihood equaios do o alwas have a soluio he maximum of he lielihood fucio aaied a he bode of he domai of paamees. So we obaied he secod ode paial deivaives of l[ q ] wih espec o paamees fo ad ad b usig MATCAD sofwae we obseved ha hese equaios give egaive values fo all > ad such ha <. Thus he desi of he ALSD is log-cocave ad hece he maximum lielihood esimaos of he paamees ad ae uique ude hese paameic esicios cf. Puig 3. I pacice oe ca esimae he paamees fo... ad of he ALSD fo paicula values of sa Applicaios Fo umeical applicaios hee we coside wo eal life daa ses of which he fis daa se is he obseved disibuio of baceial clumps pe field i a mil film ae fom Bliss ad Fishe [] ad he secod daa se is he expeimeal evidece coceig coagious disibuios i ecolog ae fom Evas [3]. We have fied he LWZD he ALSD ad he ALSD o hese daa ses ad he esuls hus obaied alog wih he coespodig values of he expeced fequecies Chi-squae saisic degees of feedom d.f. P Aaie ifomaio cieio AIC Baesia ifomaio cieio BIC ad he secod ode Aaie ifomaio cieio AICc fo which he ALSD gives he bee fi fo each of hese models ae peseed i he Tables ad. Based o he values of Chi-squae saisic P AIC BIC ad AICc fom hese ables we ca be obseve ha he ALSD gives a bee fi o he fis daa se fo 3 ad he secod daa se fo compaed o he exisig models- he LWZD ad he ALSD. 78

7 The alpha-logaihmic seies disibuio of ode Table. Obseved fequecies ad expeced fequecies of he LWZD he ALSD ad he ALSD b he mehod of maximum lielihood fo he fis daa se. No. pe ui Obseved fequec LWZD ALSD ALSD Toal ˆ.9 ˆ.5 ˆ.4 ˆ.35 ˆ.36 ˆ ω. 6 ˆ. 4 Esimaes of he paamees Chi-squae value ˆ.5 ˆ.9 ˆ. ˆ3. ˆ.5 ˆ.8 ˆ3.9 ˆ4. ˆ d.f P-value < AIC BIC AICc

8 C. Saheesh Kuma ad A. Riaz Table. Obseved fequecies ad expeced fequecies of he LWZD he ALSD ad he ALSD b he mehod of maximum lielihood fo he secod daa se. No. pe ui Obseved fequec LWZD ALSD ALSD Toal ˆ.95 ˆ.55 ˆ.6 ˆ ω. 5 ˆ. 39 Esimaes of he paamees Chi-squae value ˆ.35 ˆ.9 ˆ.5 ˆ.34 ˆ3. ˆ d.f P-value <. <..84. AIC BIC AICc Tesig of he pohesis I his secio we discuss he esig of he hpohesis : i i i m fo a paicula subse i i... i } of he se { } b usig geealized lielihood aio es ad Rao s efficie scoe es. { m 5.. Geealized Lielihood Raio Tes I case of geealized lielihood aio es he es saisic is l l l 5. l whee l l L ˆ; x i which ˆ is he maximum lielihood esimao of ; wih o * * esicios ad l l L ˆ ; i which ˆ is he maximum lielihood esimao of ude. The log- x 8

9 The alpha-logaihmic seies disibuio of ode lielihood fucio l L l L x is as defied i 3. ad he es saisic give i 5. is asmpoicall disibued as chi-squae wih m degees of feedom fo deails see Rao 4. Fo esig he sigificace of he addiioal paamees of he ALSD fo 3 i case of fis daa se we coside he followig hee ess: : : Tes : agais : 3 : 3 Tes : agais 3 : 3 3 : 3 Tes 3: agais. : I case of he secod daa se we coside he ull hpohesis as agais aleaive hpohesis : fo esig he sigificace of he addiioal paamee of he ALSD fo. We have * compued he values of l L ˆ; x l L ˆ ; x ad he es saisic give i 5. i all he above siuaios ad peseed i Table 3. * x Table 3. Compued values of l L ˆ ; x l L ˆ ; ad he geealized lielihood aio es saisic. * l L ˆ ; x l L ˆ ; x Tes saisic Daa se Tes Tes Tes Daa se d.f Chi-squae value abled value Fom able 3 i ca be obseved ha he calculaed value of he es saisic is geae ha he abled value i all he above siuaio ad hece oe ca coclude ha of he addiioal paamees ad 3 ae sigifica i case fis daa se ad he addiioal paamee is sigifica i case of secod daa se a 5% level of sigificace. 5. Rao s Efficie Scoe Tes ee he es saisic is S M φ M 5. i which M log L log L log L log L ad φ is he Fishe ifomaio maix. The es saisic S give i 5. follows chi-squae disibuio wih m degees of feedom fo deails see Rao 4. Fo esig he sigificace of he addiioal paamees of he ALSD fo 3 i case of he fis daa se we coside he followig hee ess: 8

10 C. Saheesh Kuma ad A. Riaz : : Tes : agais : 3 : 3 Tes : agais 3 : 3 3 : 3 Tes 3: agais. Fo esig of he sigificace of he addiioal paamee of he ALSD fo i case of he secod daa se we coside he ull hpohesis as : agais he aleaive hpohesis :. We have compued he values of S fo he ALSD i Tes as S i Tes as S i Tes 3 as S 3 of he ALSD i he case of fis daa se ad S 4 as i he case of secod daa se ad peseed hem i Table 4. The compuaioal deails of S i fo i 3 ad 4 ae give i Appedix A. Table 4. Compued values of Rao s efficie scoe es saisic. Tes saisic d.f Chi-squae value abled value Daa se S S S Daa se S Fom able 4 i ca be obseved ha he calculaed value of he es saisic is geae ha he abled value i all he above siuaios ad hece oe ca coclude ha he addiioal paamees ad 3 ae sigifica i case fis daa se ad he addiioal paamee is sigifica i case of he secod daa se a 5% level of sigificace. 6. Simulaio I is quie difficul o assess he heoeical pefomace of he esimaos of diffee paamees of he ALSD obaied b he mehod of maximum lielihood. So i his secio we have aemped a simulaio sud fo compaig he pefomace of he esimaos. We have simulaed hee daa ses of sample sizes 5 3 ad 6 i case of boh he ove-dispesed ad ude-dispesed siuaios of he ALSD fo 3 ad 4 b usig Maov Chai Moe Calo MCMC simulaio pocedue ad cosideed eplicaios i each case. The iiial value of he paamees assumed fo simulaig he daa ses accodig o he aue of dispesio ad he compued he values of bias ad sadad eos i case of each esimaos ae summaized i Tables 5 6 ad 7. Fom hese ables i ca be obseved ha boh he bias ad sadad eos of he esimaos of he paamees ae i deceasig ode as he sample size iceases. 8

11 The alpha-logaihmic seies disibuio of ode Table 5. Bias ad sadad eo wihi paehesis of he esimaos of he paamees ad of he ALSD fo coespodig o he paamee se: fo i ove-dispesio ad ii ude-dispesio. Paamee se Sample size Maximum lielihood esimaio ˆ ˆ ˆ i ii Table 6. Bias ad sadad eos wihi paehesis of he esimaos of he paamees ad of he ALSD fo 3 coespodig o he paamee se: i ove-dispesio ad ii ude-dispesio. Paamee se Sample size Maximum lielihood esimaio ˆ ˆ 3 ˆ i ii

12 C. Saheesh Kuma ad A. Riaz Table 7. Bias ad sadad eos wihi paehesis of he esimaos of he paamees 3 4 ad of he ALSD fo 4 coespodig o he paamees se: i ove-dispesio ad ii ude-dispesio. Paamee se i ii Sample size ˆ Maximum lielihood esimaio: ˆ ˆ ˆ ˆ Appedix. A We have compued he values of he Rao s efficie scoe es saisic coespodig o he ess descibed i secio 5. as follows: S S S S

13 Refeeces The alpha-logaihmic seies disibuio of ode [] Bliss C. I. ad Fishe R. A Fiig he egaive biomial disibuio o biological daa. Biomeics [] Chafield C. Ehebeg A. S. C. ad Goodhad G Pogess o a simplified model of saioa puchasig behavio wih discussio. oual of he Roal Saisical Socie Seies A [3] Evas D. A Expeimeal evidece coceig coagious disibuio i ecolog. Biomeia [4] Fishe R. A. Cobe A. S. ad Williams C. B. 943.The elaio bewee he umbe of species ad he umbe of idividuals i a adom sample of a aimal populaio. oual of Aimal Ecolog [5] ai G. C. ad Gupa R. P A logaihmic seies pe disibuio. Tabaios de Esadisica [6] ohso N. L. Kemp A.W. ad Koz S. 5. Uivaiae Discee Disibuios. Wile New Yo. [7] Kempo R. A A geealized fom Fishe s logaihmic seies. Biomeia [8] Khag T. F. ad Og S.. 7. A ew geealizaio of he logaihmic disibuio aisig fom he ivese iomial disibuio. Commuicaio i Saisics Theo ad Mehods [9] Khai C. G. 96. O he disibuio obaied b vaig he umbe of ails i a biomial disibuio. Aals of he Isiue of Saisical Mahemaics [] Kuma C. S. 9. A class of discee disibuios of ode. oual of Saisical Theo ad Pacice [] Kuma C. S.. Biomial Poisso disibuio evisied. Ecoomic Quali Cool [] Kuma C. S. ad Nai B. U. 3.O sueig hpe Poisso ad is popeies. Si Laa oual of Applied Saisics [3] Kuma C. S. ad Nai B. U. 3a. Ode vesio of he aleaive hpe Poisso disibuio. Ecoomic Quali Cool [4] Kuma C. S. ad Riaz A. 3. O esimaig he paamees of a exeded logaihmic seies disibuio ad is applicaios. Commuicaios fo Saisical Applicaios ad Mehods [5] Kuma C. S. ad Riaz A. 3a. A modified vesio of logaihmic seies disibuio ad is applicaios. Commuicaio i Saisics Theo ad Mehods acceped fo publicaio. [6] Kuma C. S. ad Riaz A. 4. A aleaive o he logaihmic-wih-zeos disibuio ad is geealizaio: Popeies ad esimaio. Souh Afica Saisical oual [7] Kuma C. S. ad Riaz A. 4a. Logaihmic seies disibuio of ode. The Aligah oual of Saisics [8] Kuma C. S. ad Shibu D. S. 3. O ieveed sueig Poisso disibuio ad is applicaio. oual of Saisical Theo ad Pacice [9] Mahai A. M. ad aubold.. 8. Special Fucios fo Applied Scieiss. Spige New Yo. [] Og S... O a geealizaio of he log-seies disibuio. oual of Applied Saisical Sciece [] Paaeos. ad Xealai E O some disibuios aisig fom ceai geealized samplig schemes. Commuicaios i Saisics-Theo ad Mehods [] Philippou A. N The egaive biomial disibuio of ode ad some of is popeies. Biomeical oual [3] Puig P. 3. Chaaceizig addiivel closed discee models b a pope of hei MLEs wih a applicaio o geealized emie disibuio. oual of Ameica Saisical Associaio [4] Rao C. R Liea Saisical Ifeece ad is Applicaio. oh Wile New Yo. [5] Slae L Geealized pegeomeic Fucios. Cambidge Uivesi Pess Cambidge. [6] Tipahi R. C. ad Gupa R. C A geealizaio of he log-seies disibuio. Commuicaio i Saisics-Theo ad Mehods [7] Tipahi R. C. ad Gupa R. C Aohe geealizaio of he logaihmic seies ad he geomeic disibuio. Commuicaio i Saisics-Theo ad Mehods [8] Williams C. B The logaihmic seies ad is applicaio o biological poblems. oual of Ecolog [9] Williams C. B Paes i he Balace of Naue. Lodo Academic Pess. [3] Williamso E. ad Beheo 964.Tables of he logaihmic seies disibuio. Aals of Mahemaical Saisics