Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 ISSN:99-878 Ausl oul of s d Appled Sees oul home pge:.se.om he Geelzed e-geelzed Peo Dsuo M.E. H.M. Shms d E.A. Husse Depme of Ss Fuly of Commee Al-Azh Uvesy A R I C L E I N F O Ale hsoy: Reeved 3 ue 05 Aeped 5 Augus 05 Avlle ole Sepeme 05 Keyods: e-dsuo Esmo Epoeed Geelzed Peo Kumsmy Kuoss Skeess MLE Relly. A S R A C sed o he Geelzed e-geeed dsuo Codeo e l.[5 e sudy he so-lled Geelzed e-geelzed Peo dsuo fo he fs me he G-GP dsuo s odued d suded 's ludg some spel su-model lke Kum-GP dsuo -GP dsuo geelzed Peo dsuo d s epoeed. Some suul popees of he poposed dsuo e suded ludg epl epessos fo he momes he esmo of he model pmees s dsussed esme pmees y he mehod of mmum lkelhood d povde he oseved fomo m. he flely of he e model s llused h pplo o el d se. 05 AENSI Pulshe All ghs eseved. o Ce hs Ale: M.E. H.M. Shms d E.A. Husse he Geelzed e-geelzed Peo Dsuo. Aus.. s & Appl. S. 93: 696-7 05 INRODUCION he Peo dsuo s he mos popul model fo lyzg skeed d. he Peo dsuo s fs poposed y Peo 897 s model fo he dsuo of ome. I e used o epese vous ohe foms of dsuos ohe h ome d h se hum lfe. I hs plyed vey mpo ole he vesgo of y F [ > populo szes ouee of ul esoues sue-sk d usess flues. Ald e l. 983 gves eesve hsol suvey of s use he oe of ome dsuo. he fou pmees Peo geelzed Peo dsuo s odued y Adul-Fh e l. 007. he umulve dsuo fuo df of he fou pmees Peo dsuo geelzed Peo dsuo s A dom vle s sd o follo he Peo dsuo h fou pmees geelzed Peo dsuo f he poly desy fuo pdf of s s follos: f [ d 0 hee s fo > 0 he loo pmee s sle pmee d s shpe pmees. I hs oe e popose eeso of he Geelzed Peo dsuo sed o he fmly of geelzed e-geeed deoed h he pef G-GPfo sho dsuos odued y Codeo e l. 0. o mpo spel su-models e he G dsuo d he Kumsmy geeed KG dsuo eely poposed y Codeo e l. 0. of ouse he e ype I desy fuo self ses mmedely f F s ke o e he ufom umulve fuo. he lssl e dsuo my e hezed y s desy fuo f 0 3 Coespodg Auho: M.E. H Depme of Ss Fuly of Commee Al-Azh Uvesy E-ml: m.h000@yhoo.om
697 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 hee deoes he e fuo. he gmm fuo d he o pmees d e suh h > 0 Alhough hs oly o pmees he e desy ommodes vey de vey of shpes ludg fo he sdd ufom dsuo U [0. he e desy s symme f > d s umodl he â œuâ shped he <. I hs posve < d egve ske he >. ske he he fs usg of e-geeed dsuo odued y Eugee e l.[0 hey ledy deved he fs dsuo of G lss s e Noml dsuo deoe he sdd oml df d pdf y. d. espevely he X hs e oml dsuo N 0 h desy fuo f 0 [ [ < < 4 I hs oe e ome he oks of G 5 d Adul-Fh e l.[ o deve some mheml popees of e model lled he Geelzed e-geelzed Peo G-GP dsuo d ove he esmo of he model pmees v mmum lkelhood mehod hh sems fom he follog geel osuo. Geelzed e Geeed G dsuo s o vey ommo mog sss d hs ee lle eploed he leue. I my e hezed y s desy fuo d he oespodg df fo he ls fomul f f [ F [ F 5 F GG GG F I F [ [ d 6 No d e shpe pmees o hose pmee veo d hee I. deoes he omplee e o fuo. If X s dom vle h pdf 5 e e X : gg F h > 0 > 0 d > 0 F d f e he umulve d desy fuos espevely. o mpo spel su-models: Whe he fomul edued o G dsuo hh poposed y oes 004. Whe he fomul edued o Kumsmy geeed KG dsuo hh poposed y Codeo e l.[8. hs ppe s ouled s follos. I seo II e defe he G-GP dsuo d povde epsos fo s umulve d desy fuos. A ge of F GGP I[ 0 mheml popees of hs dsuo s osdeed seos III ll VII. hese lude mome d omplee momes Ve he oeffe of vo skeess d kuoss Mmum lkelhood esmo s pefomed d he oseved fomo m s deemed seo VIII. I seo IX he flely of he e model s llused h pplo o el d se. Flly some olusos e ddessed seo X. he Geelzed e-geelzed Peo: If F s he Geelzed Peo umulve dsuo h Pmee. he equo 6 yelds he G-GP umulve dsuo fo > [ [ d >. 7 0 hee d > 0 e o-egve shpe Pmees > 0 s he sle pmee d 0 I [. deoes he omplee e o fuo. he oespodg pdf d Hzd Re Fuo e: [ {[ [ [ [ > f GGP 8
698 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 { I S GGP 9 [ h GGP { [ I [ [ [ {[ 0 Shpes: Fg. : Plos of he G-GP desy fo seleed pmee vlues. Fg. : Fgues d Plos of he G-GP desy h d hh le he K-GP d -GP espevely s Spel se h seleed dffee vlues fo ohe pmees. Fg. 3: Plos of he G-GP suvvl fuo fo seleed pmee vlues.
699 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 Fg. 4: Plos of he G-GP hzd e fuo fo seleed pmee vlues. Epsos fo he umulve d desy fuos: I hs seo e ll deve some useful epsos fo he G-GP df d pdf fo evy of ooy seg Fs oe h he s ege 7 my e e s GGP F [ 0 Alevely f s ege: 0 [ GGP F ohe epsos fo 8 my e deved usg popees of epoeed dsuo epdg he oml 8 yelds: fo ege 0 > he oml heoem ll e D D 0 3 equo8e e 0 g f GGP 4 0 hees fo el o-ege < < 0 he oml heoem ll e D D 0 5 equo8e e 0 g f GGP 6 0 hee g deoes he geelzed Peo desy fuo h Pmees hus he G-GP desy fuo e epessed s fe le omo of geelzed Peo desy s ell s some of 's mheml popees e oed dely fom hose popees of geelzed Peo dsuo.
700 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 Lmg ehvo of Geelzed e Geelzed Desy d Hzd Fuos: Lemm. he lm of G-GP desy fuo s s 0 d he lm s e udefed fo0 < < lm f GGP fo 0 fo >. Poof. s esy o demose he esul fom he desy fuo 8. Lemm. he lm of G-GP Hzd fuo s s 0 d he lm s e 7 lm h GGP udefed 0 fo0< < fo fo >. 8 Poof. s sghfod o pove hs esul fom 0. Spel Dsuos: he follog ell-ko d e dsuos e spel su-models of he G-GP dsuo. he Kumsmy-Geelzed Peo Dsuo: If he Geelzed e geelzed Peo dsuo edues o f KWGP [ {[ [ [ [ > 9 hh s he Kumsmy-Geelzed Peo Fo Dsuo "KW-GP". Whee dsuofo > 0 e he pmees of he geelzed Peo dsuo. Fo e o he Epoeed Geelzed Peo dsuo. e o he Geelzed Peo e o he hee pmees Peo dsuo. Adul-Fh e l. 007. Fg. 5: Plos of he K-GP desy h dffee vlues fo ohe pmees. he Geelzed e-compoud Weull Gmm Dsuo: If he Geelzed e geelzed Peo dsuo edues o f GCWG { { { 0
Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 ISSN:99-878 Ausl oul of s d Appled Sees oul home pge:.se.om hh s he Geelzed e - Compoud Weull Gmm dsuo "G-WG". Whee > 0 e he pmees of he Compoud Weull Gmm dsuo. Fo e o he Kumsmy-Compoud Weull Gmm dsuo"kw-cwg". Fo e o he e-compoud Weull Gmm dsuo"-wg". Fo e o he Epoeed ompoud Weull Gmm dsuo. Fo e o he Compoud Weull Gmm dsuo. Adul-Fh e l. 007. he Geelzed e-compoud Rlegh Gmm Dsuo: If d f G RG he Geelzed e geelzed Peo dsuo edues o { { { hh s he Geelzed e - Compoud Rlegh Gmm dsuo "G-RG"hee > 0 e he pmees of he Compoud Rlegh Gmm dsuo. Fo e o he Kumsmy-Compoud Rlegh Gmm dsuo"kw-rg". Fo e o Coespodg Auho: M.E. H Depme of Ss Fuly of Commee Al-Azh Uvesy E-ml: m.h000@yhoo.om he e-compoud Rlegh Gmm dsuo"-rg" Fo e o he Epoeed ompoud Rlegh Gmm dsuo. Fo e o he Compoud Rlegh Gmm dsuo. Adul-Fh e l. 007. he Geelzed e-u-xii Dsuo: If he Geelzed e geelzed Peo dsuo edues o f G d 0 [ [ [ hh s he Geelzed e - u-xii dsuo "G-u-XII". Whee > 0 e he pmees of he u-xii dsuo. Fo e o he Kumsmy- u-xii [ [ > 0 dsuo"kw-u-xii". Fo e o he e- u-xii dsuo"-u-xii" e o he Epoeed u-xii dsuo. Fo e o he u-xii dsuo. Adul-Fh e l. 007. he Geelzed e-lom Dsuo: If d 0 he Geelzed e geelzed Peo dsuo edues o fgl [ {[ [ > [ [ 3 hh s he Geelzed e - Lom dsuo "G-L". Whee > 0 e he o pmees of he Lom dsuo. Fo e o he Kumsmy- Lom dsuo"kw-l". Fo e o he e - he e-geelzed Peo Dsuo: Lom dsuo"-l" fo e o he Epoeed Lom dsuo. Fo e o he o pmees Lom dsuo. Adul-Fh e l. 007.
70 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 If he Geelzed e geelzed Peo dsuo edues o f GGP [ {[ [ [ [ > 4 Whh s he e-geelzed Peo Dsuo "-GP". Whee > 0 e he pmees of he geelzed Peo dsuo. Fo e o he Epoeed Geelzed Peo dsuo. Fo e o he Geelzed Peo dsuofo e o he hee pmees Peo dsuo. Adul-Fh e l. 007. Fg. 6: Plos of he -GP desy h dffee vlues fo ohe pmees. he e-u XII Dsuo: If d 0 he Geelzed e geelzed Peo dsuo edues o f GGP [ {[ [ [ [ > 0 5 hh s he e-u XII Dsuo "-u XII". Fo e o he Epoeed u XII dsuo. Fo e o u XII dsuo. Adul-Fh e l. 007.
703 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 Fg. 7: ee of Some Deved Spel Dsuos. Popees of Geelzed e Geelzed Peo dsuo: I hs seo ll pese some of ou desy popees y follog?? e o: Momes d Iomplee momes: Momes: Hee d heefoh le X e G-GP dom vle follog 8. he h mome of X e oed fom 8 s d g X E 0 0 6 hee 0 0 I pul he me my e epessed s 0 X E 7 hee 0 0 Iomplee momes: If Y s dom vle of geelzed Peo Dsuo "GP" h Pmees pmees he he h omplee mome of Y s gve y: d y g y R M R [ [ { 0 R 8 Fom 8 e oe h M R X he R. he he h omplee mome of equl o d g R M R [ { 0 0 [ R 9 hee 0
704 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 Ve: Usg he ppope mome epessos he ve s gve s follos: V E E 0 { 30 0 hee 0 0 he oeffe of vo: he oeffe of vo s oed s: C. V. X 00 3 skeess d kuoss: he skeess d kuoss o e luled y usg he elos 3 3 E X 3E X E X E X Skeess E X kuoss 4 3 X 4E X E X 3 6E X X Plos of he skeess d kuoss fo some hoes of he pmee s fuo of d fo some hoes of he pmee s fuo of. E X 3E Esmo d fomo m: I hs seo e dsuss mmum lkelhood he -lkelhood fuo esmo d feee fo he G GP fo e s: dsuo. Le e dom 4 X 3 33 smple fom X : G GP hee e he veo of se desy model Pmees y seg [ 34 Se > he he esme vlue fo equl he fs ode ss s U e gve y dffeeg?? [. he soe veo hee he ompoes oespodg o he pmees
705 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 [ [ [ [ 35 he mmum lkelhood esmes MLEs of he pmees e he soluos of he ole equos 35 0 hh e solved evely hese soluos ll yeld he ML esmos fo d. Fo he pmees G-GP dsuo ll he seod ode devves es. hus e eque he 6 6 u oseved fomo m 36 hose elemes e gve Apped A. y solvg hs vese dspeso m hese soluos ll yeld sympo ve d ove of hese ML esmos fo d usg 38 e ppome 00 peege ofdee evls fo e deemed espevely s Empl Applos: I hs seo e use o el d ses o ompe he fs of G-GP dsuo h hose of ohe su-models.e. he GP KGP d he pe dsuo self. I eh se pmees e esmed v he MLE mehod desed Seo 6 usg he MAHCAD sofe. Fs e dese he d ses. he e epo he he MLEs d he oespodg sdd eos peheses of he pmees d e shll pply foml goodess-of-f ess o vefy hh dsuo fs ee he el d ses. We osde he Cm è -vo Mses W d Adeso Dlg A sss. he sss W d A e desed dels Che e l. 995. I geel he smlle vlues of hese sss s he ee f o he d.e shll ompe he poposed GGP dsuo d he su-models h sevel ohe lfeme dsuos e -s LLog Gmm KW dsuos hh odued y Au. Lemoe 04. Flly e pefom he
706 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 Kolmogoov-Smov K-S ss d ess. Volge d: hs d se s pevously suded y Meeke d Eso Meeke 998 pge 383. hese d epese he mes of flue d ug mes fo smple of deves fom feld-kg sudy of lge sysem. A e po me 30 us ee slled oml seve odos. o uses of flue ee oseved fo eh u h fled: he flue used y umulo of domly oug dmge fom poe-le volge spkes dug ele soms d flue used y oml podu e. Follog Meeke d Eso e ssume he d e geeed y Weull dsuo d o use hs fo he pe of ou fmly of geeed dsuos. le : he Volge d. 75 3 47 3 8 30 65 0 300 73 06 300 300 300 300 300 6 93 88 47 8 43 300 3 300 80 45 66 le : MLEs of he model pmees sdd eos pehesesd he sss W d A fo Volge d Dsuos Pmees Ss α β δ W A G-GP 4.09 7.4 7.54 0.06.34 5.04 0.00809 0.0493.7.56 0.4 0.003 6.44 0.53 -GP 85.56 3.08 -- 0.4 4.06.7 0.038 0.784 43.46 0.8 0.03 5. 0.57 K-GP -- 8.47 0.86.84 5.50 0.57 0.33.0344 -- 6.96 0.03 0.36 5.9 0.49 EGP -- -- 0.47.3 7. 4.49 0.0907 0.78747 0.05 0.0 0.09 0.08 GP -- -- --.09.5 4.889 555.98 569.64 0. 5.3 0.80 LLog 0.09 0.3 5.47 3.5 -- 0.03867 0.7763 0.7 0.45 0.003 6.44 Gmm 5.95.7 0.480 0.757 0.8 0.4 le lss he MLEs d he oespodg sdd eos peheses of he pmees of ll he models d he sss W d A fo he Volge d se e oe h he poposed GGP model peses he smlles vlues of he sss W d A d hee should e hose s he es model mog ll he dsuos o f he d se. Uesoed d "ldde Ce pes: HeeWe osde uesoed d se oespodg o emsso mes mohs of dom smple of 8 ldde e pes. hese d ee pevously suded y Lee d Wg 003 d Lemoe d Codeo 0. ldde e s dsese hh oml ells mulply hou ool he ldde. he mos ommo ype of ldde e epules he oml hsoy of he uohelum d s ko s sol ell om.he d e s follos: le 3: ldde Ce pes 0.08 0. 0.4 0.5 0.5 0.8 0.9.05.9.6.35.4.46.76.0.0.07.09.3.6.46.54.6.64.69.69.75.83.87 3.0 3.5 3.3 3.57 3.64 3.7 3.8 3.88 4.8 4.3 4.6 4.33 4.34 4.4 4.5 4.5 4.87 4.98 5.06 5.34 5.4 5.4 5.49 5.6 5.7 5.85 6.5 6.54 6.76 6.93 6.94 6.97 7.09 7.6 7.8 7.63 7.66 7.87 7.93 8.6 8.37 8.53 8.65 8.66 9.0 9. 9.47 9.74 0.0 6 0.3 4 0.6 6.6 4.7 9.9 8.0.0 3.0 7.6 3 3. 3. 9 3.8 4. 4 4.7 6 4.7 7 4.8 3 5.9 6 6.6 7. 7. 7.3 8. 9. 0..7.6 3.6 5.7 5.8 6.3 3. 34. 36.6 43.0 46. 4 79.0 5 6 3 8 3 9 3 4 5 6 6 3.36 3.36 3.48 3.5 5.09 5.7 5.3 5.3 7.3 7.39 7.59 7.6 0.7 5 le 4: MLEs of he model pmees sdd eos pehesesd he sss W d A fo ldde Ce pes Dsuos Pmees Ss α β δ W A G-GP 64.77 3.7.30 0.0 3.4 3.38 0.0054 0.0953. 5
707 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 36.03.35 0.03 0.0 53.36 0.77 -GP 3.89 54.54 -- 0.9.40 0.68 0.0469 0.666 8.63 75.0 0.05.8 0.7 K-GP -- 0.57.9 0.95 45.9 0.85 0.09357 0.633 -- 4.64 0.03 0.9 48.35.97 EGP -- -- 0.63.03 30.46 3.87 0.34048.0570 0. 0.6 7.0.6 GP -- -- --.09.5 4.889 0.9084 5.7305 0. 5.3 0.80 LLog 0.70.348 0..89 -- 0.0707 0.0 0.5668.380 4.70.0976 Gmm.75 0.5 0.988 0.798 0.3 0.0 le 4 lss he MLEs d he oespodg sdd eos peheses of he pmees of ll he models d he sss W d A fo he ldde Ce pes d se e oe h he le 5: he Kolmogoov-Smov K-S ss d. poposed GGP model peses he smlles vlues of he sss W d A d hee should e hose s he es model mog ll he dsuos o f he d se. D Model GGP GP KWGP EGP GP K S 0.89 0.933 0.64 0.49 0.68 389.79 459.73 483.5 448.34 43.7 K S 0.548 0.458 0.484 0.98 0.3 864.84 858.79 8.5 8.5 80.5 Volge d ldde Ce Coludg Remks: he ell-ko geelzed Peo dsuo s eeded y odug hee e shpe pmees hus defg he geelzed e geelzed Peo G-GP dsuo hvg ode lss of hzd e d desy fuos. hs s heved y kg?? s he sele umulve dsuo of he geelzed lss of e dsuos. A deled sudy o he mheml popees of he e dsuo s peseed. he e model ludes s spel su-models s Kumsmy geelzed Peo K-GP epoeed geelzed Peo EGP d e geelzed Peo GP dsuos. We o he ody momesd R è y eopy. he esmo of he model pmees s ppohed y mmum lkelhood d he oseved fomo m s oed. A pplo o el d se des h he f of he e model s supeo o he fs of s ppl su-models. We hope h he poposed model my e eesg fo de ge of ssl eseh. Apped : Upo dffeeg 37 he elemes of he oseved fomo m fo he pmees d e [
708 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 [ [ [ [ [ [ [ hee [ { [ hee { [ hee [ { 3 [ hee [ { 3 5 4 [ hee { 4 [ 5 { {
709 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 7 6 [ hee { 6 { 7 { [ 8 [ [ 9 hee [ { 8 [ { 9 z 0 [ hee { 0 { [ 4 3 5 [ hee
70 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 3 { { 4 5 { { [ { 36 le 6: Some Spel Dsuos. Dsuo e GP Epoeed GP e Peo "3Pmees" e u XII 0 e Lom 0 e e II 0 e Ep.-Epoel e Rylegh-Gmm e Weull-Epoel K GP - K Peo "3 Pmees" K u X II 0 K Lom 0 K eii 0 K Ep.-Epoel K Rylegh-Gmm K Weull-Epoel Geelzed Peo hee pmees Peo o pmees u XII 0 o pmees Lom 0 e II 0 Epoel-Epoel Rylegh-Gmm Weull-Epoel REFERENCES Adul-Fh A.M. E.A. Elshepy d E.A. Husse 007. "A e geelzed Peo dsuo" Ies oul De07. Aksee A. F. Fmoye d C. Lee 008. he e-peo dsuo. Sss 46: 547563. Ald.C. d S.. Pess 983. "yes feee fo Peo populos" oul of Eoomes : 87-306. Au Lemoe. 04. "he e -s dsuo"ps oul PS04-05. Col Alede M. Guss Codeo 0. Ed M. M. Oeg d ose Ms Sd 0Geelzed e-geeed Dsuos ICMA CeeDP0-05.
7 M.E. H e l 05 Ausl oul of s d Appled Sees 93 Augus 05 Pges: 696-7 Che G. N. lksh 995. A geel pupose ppome goodess-of-f es. oul of Quly ehoy 7: 546. Choulk V. d M.A. Sephes 00. Goodess-of-f fo he geelzed Peodsuo. ehomes 43: 478484. Codeo G.M. d M. de Cso 0. "A e fmly of geelzed dsuos. oul of Ssl Compuo d Smulo" 8: 883898. Eugee N. C. Lee d F. Fmoye 00. "e-oml dsuo d s pplos". Commuos Sss. heoy d Mehods 3: 4975. Fo Meov Lluk Puk 04. "smued Peo dsuo". PoS Foum 07:. Guss Codeo M.. Melo Pee. Rodgo Slv M. Luz e 0. "he Kumsmy Peo dsuo"oul of Sss Mehodoy.Xv:04.389 [s.me. oes M.C. 004. "Fmles of dsuos sg fom dsuos of ode sss".oul of Ssl.Xv:04.389 [s.me. es 3: -43. Lee E...W. Wg 003. Ssl Mehods fo Suvvl D Alyss 3 d ed.. Ne Yok: Wley. Lemoe A.. G.M. Codeo 0. A eeded Lom dsuo. Sss pp: -7. hp://d.do.og/0.080/033888.0.5689 Mhmoud E. 0. he e geelzed Peo dsuo h pplo o lfeme d. Mhems d Compues Smulo 8: 44430. Meeke W.Q. d L.A. Eso 998. Ssl Mehods fo Relly D. oh Wley Ne Yok. Nss M.M. d N.K. Nd 0. "he e Geelzed Peo Dsuo". oul of Sss Adves heoy d Applos.6 Nume ½: -7. Peo V. 897. "ous de'oom polque" Lusee d Ps Rge d Ce.