THE TRANSMUTED GENERALIZED PARETO DISTRIBUTION. STATISTICAL INFERENCE AND SIMULATION RESULTS

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Adele Townsend
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1 Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry Collto PROQUST Advd Thologs & Arosp TH TRASMUTD GRALIZD PARTO DISTRIBUTIO STATISTICAL IFRC AD SIMULATIO RSULTS Romă TRADAFIR Vsl PRDA Sor DMTRIU Io MIRLUS-MAZILU 4 Dprtmt of Mthmts d Computr S Thl Uvrsty of Cvl grg Buhrst Fulty of Mthmts d Computr S Uvrsty of Buhrst Dprtmt of Struturl Mhs Thl Uvrsty of Cvl grg Buhrst 4 Dprtmt of Mthmts d Computr S Thl Uvrsty of Cvl grg Buhrst Astrt: I ths ppr for th Trsmutd Grlzd Prto Dstruto trodud [4] stmt th dstruto prmtrs usg th mthod of momts mmum lklhood mthod d Mthod of prolty-ghtd momts For dffrt vlus of prmtrs grt smpls of volum d dtrm from ths smpls th m d stdrd dvto omprg thm to th thortl W study th prform of th stmtg mthods usd osdrg th s d th root m squr rror d olud tht r dqut For th osoldtd prstto of th sujt pprohd th ppr ots mportt prt of th ppr [4] Som mthmtl proprts of th dstruto r prstd ths ppr Mthmts Sujt Clssfto : P 5 Ky ords: trsmutd prolty dstrutos qudrt rk trsmutto mp dstruto of ordr sttsts stmtg of prmtrs umrl smulto Itroduto I rt yrs thr hv osdrl fforts fdg sttstl modls ot ssrly symmtrl to rprst rl orld phom Gv tht my of ths phom r ot symmtrl th fforts r drtd tords skd dstrutos from othr populr dstrutos symmtrl or ot Asymmtrl pttrs tht prss dffrt dgrs of symmtry r usful tool modlg rl orld phom Strtg from symmtrl dstruto th umultv dstruto futo G d prolty dsty futo g Azzl [] proposs symmtr dstruto hos prolty dsty futo s f g G β hr β s th prmtr th symmtry Sh d Bukly [] vstgt ovl thqu for trodug skss or kourtoss to symtr or othr dstruto Aryl d Tsokos [] us qudrt rk trsmutto mp to grt fll fmly of prolty dstruto strtg from trm vlu dstruto d grlz th to prmtr Wull dstruto [] Mrov [9] Mrov d ltl [8] lt d lgrhy [4] M Mrov d Puk [] grlz dffrt kd of Ldly dstruto d Prto dstruto usg qudrt rk trsmutto mp otg dstrutos th ppltos rllty ltl t l [5] osdr lk s dstruto lr potl dstruto d y qudrt rk trsmutto mp ot trsmutd grlzd lr potl dstruto hh us modlg of lf tm dt Kh d 4 Kg [] trodu trsmutd modfd Wull dstruto s mportt ompttv modl for lf tm dstrutos Th trsmutd ddtv Wull dstruto trodud y ltl d Aryl [] usd to modl lftm dt Th purpos of ths ppr s to vstgt prolty dstruto tht otd from symmtr dstruto mly grlzd Prto dstruto d tht usd for modlg d lyzg rl-orld dt Qudrt rk trsmutto mp Dfto [] A futol omposto of th umultv dstruto futo of o prolty dstruto F th th vrs umultv dstruto futo of othr G u R GF F G u s lld th trsmutto mp hr G s osdrd s th s dstruto d F s th modultd dstruto Ovously o lso df mutul rprstto u R FG G F u thus otg pr of rk trsmutto mps ot tht th vrs umultv dstruto futo lso ko s qutl futo s dfd s F y f R { F y y [ ] } Th futos R u FG d R u GF oth mp th ut trvl [ ] I to tslf d udr

2 Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry Collto PROQUST Advd Thologs & Arosp sutl ssumptos r mutul vrss d thy stsfy R FG d R GF Dfto [] A Qudrt Rk Trsmutto Mp QRTM s dfd s 4 u R u u u u FG from hh t follos tht th df's stsfy th rltoshp 5 G G F hh o dffrtto ylds f g [ G ] hr g d f r th orrspodg pdfs ssotd th df G d F rsptvly Th fft of th QRTM s to trodu sk to symmtr s dstruto Thr s o spf rqurmt tht th s dstruto G symmtr Hovr f th G dstruto s G { < d } d hr s loto prmtr > s sl prmtr s shp prmtr g symmtr out th org th ss tht G G - hv th rsult tht th dstruto of th squr of th trsmutd rdom vrl s dtl to tht of th dstruto of th squr of th orgl rdom vrl A osqu of ths s tht f th orgl dstruto s symmtr th th QRTM prsrvs ll v momts [] Trsmutd grlzd Prto dstruto Dfto [] Lt rdom vrl dfd s 7 Y hr r prmtrs d Y ~ > p rdom vrl th th stdrd potl dstruto If s th thrshold or lor oud of th th dstruto of s th -prmtr grlzd Prto dstruto gv y { < d } d or > d 8 Th dsty prolty futo of rdom vrl s gv y or > d Dfto 4 A rdom vrl s sd to hv trsmutd grlzd Prto prolty dstruto th prmtrs R > > d f ts df s gv y F for { < d } or > d d hr s th trsmutd prmtr Th dsty prolty futo of rdom vrl s gv y f > > 9 4

3 Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry Collto PROQUST Advd Thologs & Arosp ot tht th grlzd Prto dstruto s spl s for of th trsmutd grlzd Prto dstruto Sttst proprts Qutls Th qutl q of trsmutd grlzd Prto dstruto s th rl soluto of th quto F q Th md otd y sttg 5 q 4q for 4q q l for < q < q Rdom umr grto Usg th mthod of vrso grt rdom umrs from trsmutd grlzd Prto prolty dstruto rplg qutos d q y U ~ > U Momts Th momts of trsmutd grlzd Prto dstrutd rdom vrl r gv y th follog proposto Proposto Th momts [ ] of trsmutd grlzd Prto dstrutd rdom vrl r gv s for > [ ] [ ] [ ] d [ ] [ ] 4 8 for 9 [ ] Ordr sttsts I sttsts th kth ordr sttst of sttstl smpl s qul to ts kth smllst vlu Togthr th rk sttsts ordr sttsts r mog th most fudmtl tools o-prmtr sttsts d fr W ko tht f dots th ordr sttsts of rdom smpl from otuous populto th df F d pdf f th th pdf of j s gv y 4 4

4 Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry Collto PROQUST Advd Thologs & Arosp [ ] [ ] j j F F f! j! j! f j 5 It s usful ppltos th pdf of lrgst ordr sttst gv y f for d f 5 for It s lso usful ppltos th pdf of smllst ordr sttst gv y f 7 for d f 7 for stmto of prmtrs Mthod of momts MOM I th s of > lt smpl of rdom vrl to dtrm th vlus for vtor must solv th systm [ ] [ ] [ ] 8 hr [ ] r gv y Proposto d r mprl momts of ordr Usg umrl mthod solv th systm 8 osdrg m s ths dstruto modls th dg vlus of rl phomo For smpl sz grtd y th vrso mthod th th MOM otd stmtors d th orrspodg momts r gv th t tl: Prmtr vtor Thortl ptto Smpl m Thortl vr Smpl vr ĉ â

5 Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry Collto PROQUST Advd Thologs & Arosp W osdr th prmtr 4S S 5 5 hr S From th systm 5 ot th stmtors of prmtrs g otd from th lst quto of prvous systm ftr rplg d dpdg o Cosdrg smpl sz grtd y th vrso mthod th th MOM otd stmtors d th orrspodg momts r gv th t tl: Thortl Smpl Thortl Smpl Prmtr vtor ptto m vr vr MOM 494 ĉ MOM 5 ĉ To ot th MOM stmtors o lso solv th systm [ ] Vr[ ] S Sk γ 8 Whr S d γ r mprl vr d skss rsptvly [ 5 ] S γ 8 Somtms t s usful prtl ppltos kurtoss K [ 5 ] Mthod of prolty-ghtd momts PWM Cosdr th r th prolty-ghtd momts W r gv y r [ F F ] Wr r r C hr r W ppromt Wr y r r C d ot th systm 9 [7] hr s th slto volum d dots th ordr sttsts of rdom smpl For 45

6 Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry Collto PROQUST Advd Thologs & Arosp Solvg th systm 9 ot ĉ stmtg vtor for ĉ d s th soluto of quto Cosdrg smpl sz grtd y th vrso mthod th th PWM otd stmtors d th orrspodg momts r gv th t tl: Prmtr vtor Thortl ptto Smpl m Thortl vr Smpl vr ĉ PWM ĉ PWM Mthod of mmum lklhood stmto ML W osdr th lklhood futo for f ; L d ts logrthm ; l ; L L To ot th ĉ ML stmtg ML vtor for o must solv th systm ; ; ; L L L : Cosdrg smpl sz grtd y th vrso mthod th th ML otd stmtors d th orrspodg momts r gv th t tl: 4

7 Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry Collto PROQUST Advd Thologs & Arosp Thortl Smpl Thortl Smpl Prmtr vtor ptto m vr vr ĉ ML ML 4 9 ĉ Lk msur of prforms of th prmts otd ov osdr to prform ds [ ] mly stdrd s [ ] d s umr of rus For 5 - Cs of > m [ s d root m squr rror RMS ] hr th vlus of prform ds r gv th t tls: Mthod [ ] s RMS [ â ] [ ] [ ] MOM Cs of Mthod [ ] s RMS [ ] [ ĉ ] [ ] MOM PWM ML Th rsults from th prvous tls rgrdg th s d RMS of prmtr stmtors ttl us to lv tht th proposd mthods for stmtg ths prmtrs r sutl COCLUSIOS W hv dtrmd th dstruto prmtrs of Trsmutd Grlzd Prto Dstruto usg th mthod of momts mthod of prolty-ghtd momts d Mthod of mmum lklhood of th smultd vlus W studd prform osdrg to dtors mly s d root m squr stdrd rror for dffrt vlus of prmtrs d oludd tht th proposd mthods for stmtg ths prmtrs r sutl BIBLIOGRAPHY: [] Aryl R G Tsokos C P: O th Trsmutd trm Vlu Dstruto th Applto olr Alyss [] Aryl R G Tsokos C P: Trsmutd Wull Dstruto: A Grlzd of th Wull Prolty Dstruto urop Jourl of Pur d Appld Mthmts [] Azzl A: Rfrs o th sk-orml dstruto d rltd os O-l t [4] lt I & lgrhy M: Trsmutd Qus Ldly Dstruto : A Grlzto of th Qus Ldly Dstruto Itrtol Jourl of Pur d Appld Ss d Thology [5] lt I D L S & Adul Alm A: Trsmutd Grlzd Lr potl Dstruto of th Qus Ldly Dstruto Itrtol Jourl of Computr Applto [] ltl I & Aryl G: O th Trsmutd Addtv Wull Dstruto Austr Jourl of Sttsts 4 7- [7] Mus Ldhr J Mtls C& Wlls J R: Prolty Wghtd Momts Comprd th Som Trdtol Thqus stmtg Guml Prmtrs d Qutls Wt Rsour Rs

8 Mr l Btr vl Admy Stf Bullt Volum VIII 5 Issu Pulshd y Mr l Btr vl Admy Prss Costt Rom // Th jourl s dd : PROQUST STh Jourls PROQUST grg Jourls PROQUST Illustrt: Thology PROQUST Thology Jourls PROQUST Mltry Collto PROQUST Advd Thologs & Arosp [8] Mrov F & ltl I: Trsmutd Ldly-Gomtr Dstruto d ts Appltos rv:9774v[sttm] 7 Ot [9] Mrov F: Trsmutd Rylgh Dstruto Austr Jourl of Sttsts 4 - [] Mrov Fd Puk L: Trsmutd Prto Dstruto ProStt Forum [] Kh M S & Kg R: Trsmutd Modfd Wull Dstruto: A Grlzto of th Modfd Wull Prolty Dstruto urop Jourl of Pur d Appld Mthmts -88 [] Sh W T d Bukly I R: Th lhmy of prolty dstrutos: yod Ghrm-Chrlr psos d sk-kurtot-orml dstruto from rk trsmutto mp rv prprt rv: [] Sg V P & Guo H: Prmtr stmto for -Prmtr Grlzd Prto Dstruto y th Prpl of Mmum tropy POM Hydrologl Ss Jourl [4] Trdfr R & Prd V: Trsmutd Grlzd Prto Dstruto Th th Workshop of Stf Commutos Dprtmt of Mthmts d Computr S Mtr Rom ISS 7- My

A note on Kumaraswamy Fréchet distribution

A note on Kumaraswamy Fréchet distribution AENSI Jourls Austrl Jourl of Bsc d Appld Sccs ISSN:99-878 Jourl hom pg: wwwswcom A ot o Kumrswmy Frécht dstruto Md M E d 2 Ad-Eltw A R Dprtmt of Sttstcs Fculty of Commrc Zgzg Uvrsty Egypt 2 School of Busss