Globl Jourl of Pur d Applid hics. ISSN 97-768 Volu, Nubr (7), pp. 94-956 Rsrch Idi Publicios hp://www.ripublicio.co Th o Grig Fucio of h Four- Prr Grlizd F Disribuio d Rld Grlizd Disribuios Wrsoo, Di Kurisri, Widiri, usof Us d Fiz A.. Elfki Dpr of hics, Uivrsi of Lpug, Idosi. Dpr of hics, Sisics d Phsics, Collg of Ars d Scics, Qr Uivrsi, P. O. Bo 7 Doh, Qr. Absrc Th i objciv of his ricl is o show h drivio of h o grig fucio of h four-prr of grlizd (G4F) F disribuio. Through prrizio of is o grig fucio, h bhvior i rlio o svrl wll-kow grlizd disribuios is prsd. B uilizig cluri sris psio d Sirlig forul, i is show h wih prrizio of is o grig fucio, h grlizd F disribuio igh hv spcil rlioship o svrl wll-kow grlizd disribuios, such s grlizd b of h scod kid (GB), grlizd log-logisic (G4LL), d grlizd g (GG) disribuios. Kwords: o grig fucio; Grlizd b of h scod kid disribuio; Grlizd log-logisic disribuio; Grlizd g; cluri sris; Sirlig forul. INTRODUCTION As sd b svrl uhors, h grlizd F is o of h os wll-kow grlizd disribuios i probbili odlig. I fiig survivl d of crcio pis, Cipi, l. (986) discussd h propris d iu liklihood

94 Wrsoo, Di Kurisri, Widiri, usof Us & Fiz A.. Elfki ifrc of h fil of h grlizd F disribuio. Pg, l. (998) ivsigd h pplicio of h grlizd F disribuio o iur odl fro lpho pis. Klbflisch d Pric () discussd h us of h grlizd F disribuio. Th grlizd F disribuio is prrizio of ohr wll-kow grlizd disribuios, such s h grlizd b d grlizd log-logisic disribuios (Ph-Ghi d Duog 989, Sigh 989). Accordig o Cipi, l. (986) wh rdo vribl X hs four-prr grlizd F disribuio wih dgrs of frdo d, s dod b X ~ G4F(,,,) or G4F(,), h corrspodig probbili dsi fucio (PDF) of h G4F(,,,) c b wri i h followig for; f G4F () () ( ) ( ) ( ) ; () whr,,,, d Γ is g fucio. I c b show h if w l μ=l b+ (/) l(/), σ = /, d B(,) h is b fucio, h h PDF of h G4F disribuio bcos f GB () B(, b ) b which is h PDF of h GB disribuio s dlibrd b cdold (984), d cdold d Richrds (987). If w l μ=(- - l (/))(/α) d σ = / α, whr d, h h PDF of h G4F disribuio c b wri i h for f G4LL () B l, l which is h PDF of h G4LL disribuio cosidrd b Sigh, l.(997).

Th o Grig Fucio of h Four-Prr Grlizd 94 orovr, s od b svrl uhors (Cipi, l. 986; Pg l. 998; Klbflisch d Pric ; d Zhou l. ), h grlizd F cois svrl cool usd disribuios s spcil css or liiig disribuios, such s poil, g, logorl, d Wibull disribuios. Rcl, Co (8) hs likd h G4F disribuio o h hr-prr grlizd (GG) d h hrprr grlizd log-logisic (GLL) disribuios. To lik h G4F disribuio o ohr grlizd disribuios, h hs prrizd h probbili dsi fucio (PDF) of h G4F disribuio. Howvr, i r of o grig fucio (GF), flibiliis of h G4F disribuio d rlio o ohr grlizd disribuios r pprl o oo rdil ccsibl i ll of bov lirurs. Bsd o h GF, lriv hod could b grd for copuig os d chrcrizig disribuios fro hir GFs. Bsd o GFs, Wrsoo (9) dosrd h rlioship bw GB d GG disribuios, d Wrsoo () likd h G4LL disribuio o h GB d GG disrbuio. I dvlopig ohr spcs of h G4F disribuio, i ss worhwhil o d ll of bov dvors d o rcord h i his ppr. Thrfor, h i objciv of his ppr is o driv h GF of h grlizd F disribuio d rl ohr grlizd disribuios, such s grlizd b, grlizd log-logisic, d grlizd g. Th rs of h ricl is oulid s follows. I Scio, h plici o grig fucio of h G4F disribuio is drivd. B usig prrizio of h o grig fucio of h h G4F, Scio provids discussios of h rlio bw G4F disribuio d GB d G4LL disribuios. Scio 4 cois dscripio of liiig bhvior of h o grig fucio of h G4F disribuio s grl cs of h o grig fucio of GG d GF disribuios. Fill, so cocludig rrks r od i scio 5.. OENT GENERATING FUNCTION OF THE GENERALIZED F DISTRIBUTION I r of b fucio h pdf of GF disribuio i quio () c b rwri s: g() = B(, ) ; () whr,,, d, Γ is g fucio.

944 Wrsoo, Di Kurisri, Widiri, usof Us & Fiz A.. Elfki Thor. L X b rdo vribl of h G4F(,,,) disribuio, h o grig fucio (GF) of X is giv b 4F G () Proof: d f E () X 4F G d ) B(, B lgbr ipulio w fid h followig quio d B, () B lig w rwri h quio () i h followig for d B, () 4F G d B, (4) kig us of wll-kow propr of cluri sris of h fucio, h quio (4) is giv b d! B, () 4F G d...!! B,

Th o Grig Fucio of h Four-Prr Grlizd 945 d...!! B, d d B, d! d! d d B,... d! d! d d B,... d! d!, B B, B,, B!, B!

946 Wrsoo, Di Kurisri, Widiri, usof Us & Fiz A.. Elfki... B,, B! B,, B!!!! Thrfor, h GF of h G4F disribuio is G4F! () (5) I is wll-kow h h Sirlig s pproiio forul of h g fucio (Spigl 968) is giv b ~ b z z z b z,,!,, B B B B

Th o Grig Fucio of h Four-Prr Grlizd 947 B Sirlig s pproiio, quio (5) c b prssd s G4F....! () G4F! () G4F! () (6) B cluri sris, quio (6) c b wri s 4F G () 4F G ().. o Grig Fucio (GF) of Thr-Prr GF Disribuio Bsd o h g disribuio (lik 967, d Dr 98) d grlizd b wih hr- prr disribuio (Ph-Ghi d Duog 989) obid hrprrs of grlizd F disribuio wih h followig pdf,,, d; ) B(, () f F G Thor. L X b rdo vribl of h GF (α,,) disribuio, h o grig fucio (GF) of X is giv b GF! ()

948 Wrsoo, Di Kurisri, Widiri, usof Us & Fiz A.. Elfki Proof: d f E () X F G d ) B(, B lgbr ipulio w fid h followig quio d, B (7) B lig w rwri h quio (7) i h followig for d B, X d B, (8) kig us of wll-kow propr of cluri sris of h fucio, h quio (8) c b wri s d! B, d...!! B, d...!! B, d d B, d! d!

Th o Grig Fucio of h Four-Prr Grlizd 949 d d B,... d! d! d d B,... d! d!, B B, B,, B!, B!... B,, B! B,, B! B,, B B,, B!!

95 Wrsoo, Di Kurisri, Widiri, usof Us & Fiz A.. Elfki!! Thrfor, h GF of h GF disribuio is ()! (9). THE RELATION OF THE GF DISTRIBUTION WITH GB DISTRIBUTION Wrsoo () hs hicll drivd h GF of h GB disribuio usig GF dfiio. Bsd o rprrizio of h GF of h GLF disribuio, us pssiv sc- Th GF of h GB disribuio is providd i his scio. Th rprrizio proposiio is sd d provd. Proposiio. L X b rdo vribl hvig h G4F (µ, σ,, ) o d l b l d, h X hs h GB(, b,, ) o. Proof: G4F! l lb!

Th o Grig Fucio of h Four-Prr Grlizd 95 l lb!! b! b This is h o grig fucio of h GB (,b,, ) sd b Wrsoo (). 4. THE RELATION OF THE GF DISTRIBUTION WITH GLL DISTRIBUTION Wrsoo () drivd hicll h GF of h GLL disribuio usig dfiiio of GF s. Bsd o rprrizio of h GF of h GF disribuio, h GF of h GLL disribuio is providd i his scio. Proposiio 4. L X b rdo vribl hvig h GF (µ, σ,, ) o d l d, h X hs h GLL(,,, ) o. Proof: G4F! l!

95 Wrsoo, Di Kurisri, Widiri, usof Us & Fiz A.. Elfki l! l!!! 5. THE RELATION OF THE G4F DISTRIBUTION WITH GF DISTRIBUTION Equio (8) hs hicll drivd h GF of h GF disribuio. Bsd o rprrizio of h GF of h GF disribuio, h GF of h GF disribuio is providd i his scio. Proposiio 5. L X b rdo vribl hvig h G4F (µ, σ,, ) o d l l d, h X hs h GF(,, ) o. Proof: G4F!

Th o Grig Fucio of h Four-Prr Grlizd 95 l l! l l! l!! This is h o grig fucio of h GF(α,, ) sd i quio (9). 6. THE OENT OF THE GG AS A LIITING OENT OF THE GF DISTRIBUTIONS I his scio, proposiios of liiig o propris of h GF disribuio r sd d provd. Proposiio 6. Th GF(µ,σ,,) disribuio covrgs o h GG disribuio s ds o d l l, d. Proof: G4F! li li

954 Wrsoo, Di Kurisri, Widiri, usof Us & Fiz A.. Elfki li l l! li! li l l! li! li li li li!! li li

Th o Grig Fucio of h Four-Prr Grlizd 955! li! li Usig Sirlig s pproiio forul of h g fucio, h liiig o propr of h G4F(µ,σ,,) disribuio c b wri s: li X GF l l,,, li li!!! li... This rsul is h GF of h GG sd b Wrsoo(). Thus, h G4F disribuio covrgs o h GG disribuio s ds o d, d l. 7. CONCLUSION Th o of h grlizd F disribuio is prrizio of h grlizd b of h scod kid (GB) d h grlizd log-logisic (GLL). Th o of h grlizd g (GG) disribuio is h liiig o of (GF) disribuio. orovr, sic h os of h g d poil disribuios r spcil css of h o of h grlizd g disribuio (Wrsoo 9), h os of boh spcil disribuios r lso spcil css of h os of h o of h GF disribuio.

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