Joua of Sascs: Advacs Thoy ad Appcaos Voum 8 Num 7 Pags 43-64 Avaa a hp://scfcadvacsco DOI: hp://dxdoog/864/saa_79 ON STATISTICAL PROPERTIES OF THE EXPONENTIATED TRANSMUTED INVERTED WEIBULL DISTRIBUTION OGUNDE ADEBISI ADE OMOSIGHO DONATUS OSARETIN ad FATOKI OLAYODE Dpam of Mahmacs ad Sascs Th Fda Poychc Ado-E Nga -ma: dz95@yahoocom do@yahoocou Dpam of Sascs Ogu Sa Isu of Tchoogy Igsa Ogu Sa Nga -ma: fao_oayod@yahoocom Asac Ths wo oducs a w gazao of h o paam vd Wu dsuo Th quadac a asmuao appoach has vsgad Ths w dsuo s amd xpoad asmud vd Wu (ETIW) dsuo whch s fx ad capa of modg vaous shaps of agg ad fau chaacscs Th pops of h w mod a dscussd ad h maxmum hood smao s usd o sma h paams Expc xpssos w dvd fo h qua mom ad od sascs w xamd Mahmacs Suc Cassfcao: 7-XEX 65EXX Kywods ad phass: xpoad asmuao map maxmum hood smao qua Rcvd Novm 4 7 7 Scfc Advacs Pushs
44 OGUNDE ADEBISI ADE a Ioduco I aayzg fm daa vd Wu dsuo s o of h mos popua poay dsuo o modg h f m daa wh som mooo fau as I [] Fa a sudd h pops of h vd Wu dsuo ad s appcao o fau daa I [7] Ogud a vsgas h pops of h asmud vd Wu dsuo I [] Mudhoa a oducd h xpoad Wu dsuo as gazao of h sadad Wu dsuo ad appd h w dsuo as a sua mod o h us-moo fau m daa I [] Kha a xpad h fxy of h h paams vd Wu dsuo ad s sd pops I [3] Mudhoa ad Huso vwd h xpoad Wu dsuo wh w masus I [4] Aya ad Tsoos sudd h pops of asmud Wu dsuo I [6] Movc poposd ad sudd h vaous sucua pops of h asmud Raygh dsuo I [7] Kha ad Kg oducd h asmud modfd Wu dsuo Tasmud Lomax dsuo s psd y Ashou ad Ehwy [8] I [9] Movc ad Pua oducs asmud Pao dsuo Tasmud gazd a xpoa dsuo oducd y Eaa a [] I [8] Hady Eahm xamd h pops of h xpoad asmud Wu dsuo as a gazao of h Wu dsuo Expoad Tasmud Ivd Wu (ETIW) Dsuo A adom vaa T s sad o hav a o paam vd Wu dsuo wh paam > f s cumuav dsy fuco (cdf) s gv y G() ()
ON STATISTICAL PROPERTIES OF THE 45 ad h poay dsy fuco s gv g () x () A adom vaa T s sad o hav a asmud dsuo f s cumuav dsy fuco (cdf) W s gv y W () ( ) G() [ G( x) ] (3) wh G () s h cdf of h as dsuo fuco Th cdf F () of h xpoad asmud dsuo s gv y F () W () {( ) G() [ G( x) ] } (4) F () Bu f () hfo dffag Equao (4) h w hav () ()[ ()] g g G { G() }{ ( G( ) } (5) Comg Equaos () ad (4) gv h cdf of h ETIW dsuo as: () ( ) F (6) Aso comg Equaos () () ad (5) w yd h pdf of h ETIW dsuo as () f (7) wh > > a h shap paams ad s h asmud paam Usg ss psao of Pudov a [6] gv as ( x) ( ) x ( ) (8)
46 OGUNDE ADEBISI ADE a Equao (6) w asfom o f () ( ) (9) Usg h gazd oma xpaso gv as ( z) ( ) Z () Equao (9) w yd () f ( ) ( ) ( ) () Th gaph ow dpcs h havou of h pdf of ETIW dsuo a dff paams vaus wh a ad c Fgu Th gaph of h pdf of ETIW dsuo
ON STATISTICAL PROPERTIES OF THE 47 3 Mxu Rpsao of h Dsuo Usg h gazd oma xpaso gv Equao () W ca w Equaos (4) ad (5) as F () [ G( x) ] [ G( x) ] () Fuh smpfcao gvs F () [ G( x) ] ( ) [ G( x) ] [ G( x) ] (3) Fay w hav F () ( ) [ G( x) ] (4) Th cdf of h ETIW dsuo ca xpssd as a mxu dsy as () ( ) F (5) Aso fo h pdf w hav () ( ) f (6) 4 Sasca Pops Ths sco xamd h sasca pops of ETIW dsuo whch cuds h qua mda h o-ca mom ad h mom gag fuco
48 OGUNDE ADEBISI ADE a 4 Qua fuco Th U -h qua fuco u of h ETIW dsuo s h a souo of h quao Th w hav F ( u ) u (7) ( ) u (8) O smpfyg Equao (8) ad aso y quag x w hav u x( ) x (9) Sovg h aov quao usg quadac fomua w yd h qua fuco of ETIW gv as ( ) 4 u u () Th mda pacua ca oad y ag h vau of u 5 h ( ) 4( 5) 5 () A xpsso fo h ow quas upp qua ca aso oad y ag h vau of u o 5 75 spcvy
ON STATISTICAL PROPERTIES OF THE 49 4 Radom um gao Usg h mhod of vso adom ums fom ETIW dsuo ca gad wh q ~ U( ) as h souo of h quao q ( ) Ths gvs ( ) 4 q () 43 Moms ma vaac swss ad uoss of ETIW dsuo Th -h od moms fo ETIW dsuo ca oad as foows fo a adom vaa T: Isg Equao (7) (3) w hav E( T ) E( T ) f d (3) Expadg Equao (4) ad spg o h w oa ( ) d (4) z ( ) d ( ) (5)
OGUNDE ADEBISI ADE a 5 d z (6) 3 d z (7) Sovg fo z z ad 3 z y g u Equaos (5) (6) ad w Equao (7) w hav du u z u (8) du u z u (9) 3 dw w z w (3) Usg a gazd gamma fuco o summaz Equaos (8) (9) ad (3) wh dw w w (3) Th w hav z (3) z (33)
ON STATISTICAL PROPERTIES OF THE 5 3 z (34) Th comg Equaos (8) (9) ad (3) w hav h h - mom of h ETIW dsuo gv as T E (35) Usg Equao (3) w oa h d d s 3 ad h 4 mom fo 3 4 w hav µ (36) µ (37) µ 3 3 3 3 (38)
5 OGUNDE ADEBISI ADE a 4 4 ( ) µ ( ) 4 ( ) 4 (39) Th ma of ETIW dsuo s h fs mom aou h og ( µ ) whch cospods o Equao (3) I h foows ha h vaac ( µ ) h coffc of vaao ( ρ ) h coffc of swss ( γ ) ad h coffc of uoss ( γ ) of ETIW dsuo a spcvy oad as µ µ ( µ ) (4) µ µ ( µ ) ρ (4) µ µ µ 3 µ 3 3µ µ ( µ ) γ (4) 3 3 ( µ ) µ ( µ ) [ ] µ 4 µ 4 4µ 3µ 6µ µ 3( µ ) γ (43) ( µ ) µ ( µ ) [ ] Th mom gag fuco of h ETIW dsuo s gv y T M () E( ) E( T ) (44) Susug Equao (35) o Equao (44) w hav ( ) M () ( ) ( ) ( ) ( ) ( ) (45)
ON STATISTICAL PROPERTIES OF THE 53 5 Ray Aayss I hs sco w oad a xpsso fo h suvva aayss hazad a h cumuav hazad a ad h ma sdua f fuco fo h asmud xpoad vd Wu dsuo 5 Th suvva fuco Th asmud xpoad vd Wu dsuo povds a usfu oo fo modg fm daa aayss fo a gv sysm Th suvva fuco of ETIW dsuo ca oad fom h ao gv as R F (46) Pug Equao (6) (4) w oa () ( ) R (47) Th fgu ow usas h havou of h suvva fuco of h ETIW dsuo fo som scd vaus of h paams Fgu Th gaph of h suvva fuco of ETIW dsuo
54 OGUNDE ADEBISI ADE a Fgu Th gaph of h suvva fuco of ETIW dsuo 5 Th hazad a fuco Th hazad a fuco s aoh mpoa chaacsc of s ay masum I ca oad y f h () R() (48) Pug Equaos (7) ad (47) (48) w hav x h () (49) ( ) Th gaph ow dpcs vaous shap of hazad fuco of h ETIW dsuo fo vaous vaus of h paams hs dmosas s capay modg dff fau phoma
ON STATISTICAL PROPERTIES OF THE 55 Fgu 3 Th gaph of h hazad fuco of ETIW dsuo Fgu 3 Th gaph of h hazad fuco of ETIW dsuo
56 OGUNDE ADEBISI ADE a Fgu 3 Th gaph of h hazad fuco of ETIW dsuo 53 Th cumuav hazad a fuco Th cumuav hazad a fuco s gv y Isg Equao (47) (5) w hav H() h() d R() (5) () ( ) H (5) wh H () s h oa um of faus o dah ov a va of m whch dscs how h s of a pacua sysm vas wh m fo ETIW dsuo 6 Ry Eopy Th Ry opy of a adom vaa T pss a masu of ucay A ag vau of opy dcas h ga ucay h daa Th Ry [4] ad Baow a [3] oducd h Ry opy dfd as
ON STATISTICAL PROPERTIES OF THE 57 ad og > d x f T Z (5) Usg h pdf (7) Equao (5) w hav og d T Z (53) Th aov quao ca xpssd as m x f (54) wh m Th Ry opy s gv y og d m T Z (55) Rpsg d Q Th w hav d Q (56)
OGUNDE ADEBISI ADE a 58 W w w hav Q (57) Fay h Ry opy of ETIW dsuo s gv as og m T Z (58) 7 Od Sascs Od sascs ma h appaac may aas of sasca hoy ad pacc L T T T a adom samp fom ETIW dsuo h pdf of h h - od sascs; say T : s gv y () : x F B x f f (59) Usg Equaos (6) ad (7) (59) w hav () B f : (6) W ca summaz h aov xpsso as () () () : h h f φ φ (6)
ON STATISTICAL PROPERTIES OF THE 59 wh φ B φ B 8 Esmao of h Paams I hs sco mhod of maxmum hood s usd o sma h paams ad aso w cosuc a cofdc va fo h uow paams H w fd h smaos fo h ETIW dsuo L T T T a adom samp fom ~ ETIW T wh osvd vaus h h hood fuco L L : ca w as L (6) Hc h og-hood fuco L coms og og og (63) Th compo of h sco vco a oad y dffag (63) wh spc o h paams ad gv as ow: og og (64)
6 OGUNDE ADEBISI ADE a ( ) (65) { } { [ ]} ( ) { } { } ( ) ( ) (66) { [ ]} W ca dv h ( δ) % cofdc vas of h paams ad y usg vaac covaac max as h foowg foms: ˆ ± Z vaˆ ˆ ± Z va ˆ ˆ δ δ ± Z δ va ˆ wh Z δ s h upp h δ pc of h sadad oma dsuo 9 Appcao W cosd a daa s of h f of fagu facu of Kva 373/poxy ha a suc o cosa pssu a h 9% sss v u a had fad so w hav comp daa wh h xac ms of fau Ths daa a: 5 886 89 5 33 345 4763 565 567 6566 6748 675 6753 7696 8375 839 845 8645 885 93 9 9836 483 596 773 733 57 766 985 3 353 355 4595 488 578 5733 783 763 746 763 7746 875 8375 853 888 8878 888 936 9558 48 48 93 93 33 46 878 33 347 353 495 56 99 356 3678 3445 34846 37433 37455 3943 4873 545 54435 5595 6554 996 Fo pvous suds wh hs daa ss s Adws ad Hzg []
ON STATISTICAL PROPERTIES OF THE 6 Ta Summay of daa o fagu facu of Kva 373/poxy a 9% sss v M Low qua mda Upp qua Ma Max Vaac Swss Kuoss Rag 5 948 736 96 959 996 4774 946 868 979 Fgu 4 Th gaph of h mpca dsy ad h cumuav dsuo fuco of h Kva 373/poxy daa
6 OGUNDE ADEBISI ADE a Ta Esmad paams of h TIWD EIWD ad IWD Mod Esmas ( θˆ ) ETIWD 65763 68 68 89463 ( ) (496) (996) (93) TIWD 774 693 5483 ( ) (3994) (575) ( ) EIWD65763 868 7588 53539 ( θ ) (88) (54) ( ) IWD 73 5478 ( ) ( ) (474) ( ) Ta 3 Masus of goodss of f Mod K-S AD W AIC BIC HQIC CAIC ETIWD 586 3647 3887 6395 65933 633 6359 TIWD 44 4983 856 38967 338 389 38967 EIWD 9 5966 35 378 3574 394 343 IWD 9 569 936 3556 3887 3487 36 W mpoy h sasca oos fo mod compaso such as Komogoov-Smov (K-S) sascs Adso Dag (AD) sasc Camm vo Msss sasc (W) Aa fomao co (AIC) Coss Aa fomao co (CAIC) Haa Qu fomao co (HQIC) ad Baysa fomao co (BIC) o choos h s poss mod fo h daa ss amog h compv mods Th sco co s ha h ows AIC CAIC BIC HQIC AD ad W sasc cospod o h s f mod Cocuso Amog h mods cosdd h s mod s h xpoad asmud vd Wu dsuo fo h wo daa ss
ON STATISTICAL PROPERTIES OF THE 63 Rfcs [] A Fa H Esaouh E Md ad M Maova Th xpoad vd Wu Dsuo App Mah If Sc 6() () 67-7 [] G S Mudhoa D K Svasava ad M Fm Th xpoad Wu famy: A aayss of h us-moo fau daa Tchomcs 37(4) (995) 436-445 [3] G S Mudhoa ad A D Huso Expoad Wu famy: Som pops ad food daa appcao Commu Sasca Thoy ad Mhod 5 (996) 35-383 [4] G R Aya ad Ch P Tsoos Tasmud Wu dsuo: A gazao of h Wu poay dsuo Euopa J Pu ad App Mah 4() () 89- [5] W T Shaw ad I R Bucy Th achmy ofpoay dsuos: Byod Gam-Cha xpasos ad a Sw-Kuoc-Noma dsuo fom a a asmuao map axv pp axv:9434 (9) [6] F Movc Tasmud Raygh dsuo Ausa J Sa 4() (3) -3 [7] M S Kha ad R Kg Tasmud modfd Wu dsuo: A gazao of h modfd Wu poay dsuo Euopa J Pu ad App Mah 6 (3) 66-88 [8] S K Ashou ad M A Ehwy Tasmud Lomax dsuo Am J App Mah Sa (6) (3) -7 [9] F Movc ad L Pua Tasmud Pao dsuo PoSa Foum 7 (4) - [] I Eaa L S Da ad N A Adu-Am Tasmud gazd a xpoa dsuo I J Comp App 83(7) (3) 9-37 [] M S Kha G R Pasha ad A H Pasha Thoca aayss of vs Wu dsuo WSEAS Tas Mah 7() (8) [] D F Adws ad A M Hzg Daa: A coco of poms fom may fds fo h sud ad sach wo Spg Ss Sascs Nw Yo 985 [3] R E Baow R H Toad ad T Fma A Baysa aayss of sss upu f of Kva 49/poxy (984) [4] A L Ry O masu o opy ad fomao I Fouh By Symposum o Mahmaca Sascs ad Poay (96) 547-56 [5] R C Gupa ad R D Gupa Popooa vsd hazad mod ad s appcaos J Sa Pa If 37() (7) 355-3536
64 OGUNDE ADEBISI ADE a [6] A P Pudov Y A Bychov ad O I Machv Igas ad Ss ( ad 3) Godo ad Bach Scc Pushs Amsda 986 [7] A A Ogud O Fao ad O I Osgha O h appcao of asmud vd Wu dsuo Goa J Sc Fo Rsach 7(6) Vso [8] Ad E Hady Eahm Expoad Tasmud Wu dsuo A gazao of h Wu dsuo Wod Acadmy of Scc Egg ad Tchoogy I J Mah Comp Naua ad Phys Eg 8(6) 4 g