The Log-Gamma-Pareto Distribution

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1 aoa Joa of Scc: Bac ad Appd Rach JSBAR SSN P & O hp:odphp?oajoaofbacadappd Th Lo-Gaa-Pao Dbo Sa Ka Aho a Dah Raha Sad b Mahod Ab Aa Fah c* a of Saca Sd & Rach Cao Uv 5 Ahd Zo S Oa Gza ad 63 p b of Saca Sd & Rach Cao Uv 5 Ahd Zo S Oa Gza ad 63 p c of Saca Sd & Rach Cao Uv 5 Ahd Zo S Oa Gza ad 63 p a a: Ahoa@hoaco b a: dah@ahooco c a: ahodaa5@ahooco Abac W d h -aa-pao dbo hch cd a pca ca o od ch a aa-pao ad Pao dbo W doa ha d fco a f a cobao of Pao d So ahaca pop of h dbo a dvd ch a o dbo of h od ac Shao ad R op ad a op chaaczao W a hood ao o a od paa ad a appcao o a a daa a poa W a ado b fo h cdf of h dbo ad oba h a ba a qa o adad o Ko ad S fo ach paa Kod:Lo-aa-ad dbo; Pao dbo; o; od ac; op odco Saca dbo a pa a v poa o h cfc ach c coz h pobab dbo of h ap d do h od a ao * Copod aho -a add: ahodaa5@ahooco 357

2 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp So f a dbo o a v daa a poo f To da h h pob a aca ap o az h dbo od o podc a b f fo h daa Th aho [] odcd o fa of dbo ad b -aa ado vaab Th o fa of dbo ha h cav dbo fco cdf a G F [ F ] d R > ad G F d R > pcv h h cop aa fco F F h vva fco of ad d h pp cop aa fco Th cdf F dbo Th aho [] av h copod o pobab d fco pdf a fd o a h pa F F f R > ad F F f R > 3 pcv h f h pdf of h pa dbo F Th o of h paa ad o coo a h ad of h dbo ad v h fb fo od fca d o ad daa Th aho [] h hod ha h fa ad 3 a azd fo of dbo of cod va ad h a ao obad b app h v pobab a afoao o h -aa dbo h o ha o o h h fa of dbo dfd b Cod h Pao dbo h cdf h aho [] F > > 4 4 qao d h -aa-pao LGP dbo fco a foo f : 358

3 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp G d 5 U h afoao qao 5 dc o G d > 6 h > ad h pp cop aa fco dfd abov 8 Lo-aa-Pao dbo fco G 5 5 G 5 5 G 5 G F : Pob hap of LGP dbo fco fo dff va of ad Th d fco copod o 6 v b f > > 7 h h cop aa fco W ova h of h dbo o a F d h Pao dbo ad h aa- Pao dbo odcd b h aho [3] Scod fo oba h dbo of h o cod va fo a qc of dpd ad dca dbd ado vaab fo a popao h h Pao dbo f a ado vaab Y foo h -aa dbo h paa ad h h ado vaab foo h LGP dbo h paa Y 359

4 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp ad f a ado vaab Y foo h aa dbo h paa ad h h ado vaab foo h LGP dbo h paa vaab hav d 7 dod b ~ LGP ad A ado Lo-aa-Pao d fco F : Pob hap of LGP d fco fo dff va of ad Th hf of h LGP dbo v b f 3 h > > Th LGP qa fco ca b obad b v 6 a Q z λ h < λ < ad z [ λ ] Th v cop aa fco aad pd o d ahaca ofa h aho [4] U qao 7 o pca ca ca b obad a foo: Fo h aa-pao GP dbo obad 36

5 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp > > hch odcd b h aho [3] 3 Lo-aa-Pao hazad a fco h 5 h 3 h 5 h F 3: Pob hap of LGP hazad a fco fo dff va of ad Fo h Pao dbo obad > > hch odcd b h aho [] Th ac oazd a foo co p 7 a a f a cobao of Pao d fco co 3 oba h o ad o a fco co 4 povd pc po fo h Shao ad R op Ao popo ab coa fo a op chaaczao of h -aa ad fa qao co 5 dc h dbo of h h od ac ad h o W a h od paa b a hood hod co 6 co 7 povd a appcao o a a daa ad a a ado b fo h LGP dbo o a h f of h od pao fo h LGP D Fco 36

6 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp Cod h d fco 7 ad Th qao 7 ca b a: ] [ U h po h oba N h boa ho ad a ad cod h o a po ad o a pov h aho [5] oba b a h b a a b ] [ ad a b W ca b ad h f b h ; f f do h d fco of h Pao dbo h hap paa ad ca paa U h a hodo of h aho [4] df } { 3 Z a a d ad h h fo b Hc ca 36

7 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp f qao 8 va ha h LGP d a a cobao of Pao d So va of ahaca pop ca b da obad fo ho of h Pao dbo 3 Mo ad Mo Ga Fco L b a ado vaab dbd accod o qao 7 ad Z b a ado vaab h Pao d fco f ~ Pao Z 3 Mo o ha h h o abo zo of Pao dbo Z Th b qao 8 h h o abo zo of h LGP dbo v b Z A a aav hod o caca oda o of h LGP dbo ho qao 8 b qao 7 oba d Ma h bo oba h h o abo zo a foo: 9 d B h pao h aho [6] qao 9 dc o d 363

8 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp d paca h f o o abo zo ca b dvd b a ad qao a ad Hc h vaac of LGP dbo ca b a obad 3 Mo Ga Fco o ha h o a fco f of h Pao dbo v b M Z Z Th f of h LGP dbo ca b v b z M A a aav hod o caca h f ho qao 8 cod!! d d M B qao qao oba h o a fco a! M 4 So op Th op p a a of ca of a ado vaab a poa cocp a fd 364

9 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp of cc ch a ho of cocao phc ad cooc h aho [7] W povd po fo h Shao op ad R op of h LGP dbo ad p ab coa fo h a op chaaczao of h -aa-ad fa 4 Shao op Th aho [7] odcd h o popa a of op hch cad Shao op Fo a coo dbo G h d h Shao op dfd a H Sh [ ] { [ ]} {[ ]} d Fo h LGP dbo h Shao op v b H Sh [ ] ψ d ψ d h h daa fco 4 R op Aoh popa a of op h R op h aho [8] dfd b R v v [ ] v d hv > ad v Ra qao 7 o h po v ad a pao o ho co oba [ ] v v v v v v b [ v ] v h v ad dfd co a h abov po fo b o ad o a a cobao of Pao o of od v h R op of hch foo h LGP dbo dc o 365

10 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp R v v v v v v v v b v v 43 Ma op Chaaczao Th aho [9] hod ha h Shao op a ao b d o df a pobabc od h aho [4] Sppo a ca of pobab d d a of coa F { [ T ] } fo hch a pcao a ad o ad b f O hod choo a b fo F a h d fco fo a ado vaab f az h Shao op Th cho d cad h a op dbo Th aho [3] povdd ab coa fo h aa-ad dbo ch ha h a op dbo q Fo h -aa-ad dbo dfd popo h foo coa: [ F ] ψ ad [ f ] [ f F ] Y Y Gaa ad ψ h ~ do h daa fco ca b ho ha d h abov coa h a op dbo q Th poof of h a v a o Zoafo ad Baaha aa-ad ca ad h od 5 Th Dbo of h h Od Sac povd h aho [6] ha h d fco of h h od ac of h G dbo a f hd of G d fco ad accod o h aho [4] h d fco of h h od ac of h GW dbo a f hd of Wb d W ha pov ha fo a -aa-ad dbo h d h d of h h od ac a ado ap of z ca b pd a a f hd of -aa-ad d dfd Cod d ado vaab dbd accod o Th pdf of h h od ac a : v b!!! : [ G ] G U h boa ho ad qao oba 366

11 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp ] [!!! : F Sc ] [ F F γ ad h po fo h aho []! γ ca p h pdf : a 3!!!! : F F Foo h aho [4]! c ad h po ad o a pov a co o ] [ ] [ F d F c h c d ] [ d c c d ad Sb fo qao qao 3 oba 4!!! : d h ad do h -aa-ad fa h o paa ad Th obad fo 4 ha h pdf of h h od ac fo a ado ap of z fo a -aa-ad fa h d ca b pd a a f of -aa-ad d qao 8 ad 4 ca b d o h pdf of h h od ac fo a ap of z of h aa-pao dbo a a a cobao of Pao d a foo: 367

12 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp !!! : f d h f Pao d h ca paa ad hap paa ad h h fo b Th h o abo zo of h h od ac fo h -aa-pao dbo v b!!! : d 6 Ma-hood ao h co cod h ao of h o paa b hod of a hood Fo a ado ap of z fo h LGP dbo h -hood fco bad o h v ado ap L Th f paa dvav of h -hood fco h pc o h paa ad a pcv L ψ 368

13 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp L L ad 5 L qa h of qao 5 h zo oba ψ ad 6 369

14 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp Th of qao 6 ca b ovd ca aca paca ad h ML ad ca b obad Ud ab a codo h dbo of h a-hood ao φ of h vco paa φ v b φ φ ~ N 4 φ h φ h Fh foao a v b φ h L ψ L L L [ ]{[ ]} L [ ]{[ ]} L [ ] L [ ] L [ ] { [ [ ] {[ ]} ]} 37

15 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp L [ ] { [ ]} [ ] {[ ]} [ [ ] ][ ] ad L [ ] [ ] { [ ]} [ [ ] {[ ]} ][ ] 7 7 Appcao Th LGP dbo copad o h aa dbo h Pao dbo h aa-pao GP dbo odcd b h aho [3] ach dbo a fd o h daa cop fac Mahcad W cacad h ML fo ach od paa ad h adad o Aa foao Co AC Baa foao Co BC ad h ah of h azd hood fo ach f W cacad h Koooov-Sov ac ad p-va o dcd hh h LGP dbo o pca ca povdd a po f Th daa d h appcao fo Cooado Ca C Cooado Sa Uv v h aho [6] Th of h daa a pd ab Fo h ca obv ha h LGP dbo povd a v ood f o h daa food b h aa GP ad Pao dbo Tab a ad av ood-of-f a fo h o a a daa Dbo a LL AC BC K-S p-va Gaa Pao GP LGP

16 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp h h va h pah h adad o of h ML' of h paa Th cod paa dvav 7 a h vaad a h d a of h LGP dbo' paa od o b d fd h foo vaac-covaac a fo h paa ad : φ Ao a ap of z 5 3 fo LGP dbo fo dff va of h paa ad popod ado b ao ad h h a hood a fo ach ap b obad a h h a ba a qa o adad o ad o of ho a fo dff ap z ab h a ba a qa o adad o ad o fo h ML of h paa ad fo ado ap of z 5 3 Th ab ho ha h a qa o ad h ba fo h paa aca paca cad Mahcad ad dca a h ap z ca W d a Tab Ma Ba Ma Sqa o Sadad o S ad Ko fo h a of LGP dbo' paa fo dff va of h paa Ma Ba MS Sadad o S Ko

17 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp Ma Ba MS Sadad o S Ko Coco W odcd ad dd h -aa-pao LGP dbo ad dvopd fo pvo paa poof o a o h -aa-ad fa h dbo of h h od ac ad a op chaaczao Th a pop of h dbo ca b obad b p d fco a a a cobao of Pao d fco v ha h pdf of a b of h -aa-ad fa of dbo ad 3 ca b pd a a a cobao of h pa pdf hch b azd h fa Th a daa ap h LGP dbo a a povd aav o h Pao dbo a a dbo of co ad ab Acod W a af fo h cobo co ad o ad b h aoo f hch povd h pao ad co of h pap W ao ha h do of h oa fo h ad ao Rfc [] M A S M T K Moafaad J Ahad "Lo-aa- ad fa of dbo"a oa of Thoca ad Appd Saco [] U J D ad M J Noohab "ffc ao h Pao dbo"saca Mhodo o 7 373

18 aoa Joa of Scc: Bac ad Appd Rach JSBAR 4 o 6 No pp [3] K Zoafoad N Baaha"O fa of ba ad azd aa-ad dbo ad aocad fc"saca Mhodo o 6 9 [4] L G B Pho G M Codo ad J S Nob"Th aa-poadwb dbo"joa of Saca Tho ad Appcaoo [5] S Gadhad M Rzh Tab of a ad podcacdc P Sa Do [6] M M Rcad N Baaha"Th aa-poad poa dbo"joa of Saca Copao ad Sao o 8 [7] C Shao"A ahaca ho of cocao"th B S Tchca Joao [8] A R"O a of op ad foao" : Pocd of h foh B po o ahaca ac ad pobab Uv of Cafoa P B 96 [9] T Ja "foao ho ad aca chac"phca Rv o [] S Nadaaah ad M Pa "pc po fo o of aa od ac" B of h Baza Mahaca Soc o

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