ON STATISTICAL PROPERTIES OF THE EXPONENTIATED TRANSMUTED INVERTED WEIBULL DISTRIBUTION

Transcription

1 Joua of Sascs: Advacs Thoy ad Appcaos Voum 8 Num 7 Pags 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 Scfc Advacs Pushs

2 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() ()

3 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)

4 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

5 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

6 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

7 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)

8 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)

9 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 (38)

10 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)

11 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

12 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

13 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

14 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

15 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)

16 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)

17 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)

18 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: Fo pvous suds wh hs daa ss s Adws ad Hzg []

19 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 Fgu 4 Th gaph of h mpca dsy ad h cumuav dsuo fuco of h Kva 373/poxy daa

20 6 OGUNDE ADEBISI ADE a Ta Esmad paams of h TIWD EIWD ad IWD Mod Esmas ( θˆ ) ETIWD ( ) (496) (996) (93) TIWD ( ) (3994) (575) ( ) EIWD ( θ ) (88) (54) ( ) IWD ( ) ( ) (474) ( ) Ta 3 Masus of goodss of f Mod K-S AD W AIC BIC HQIC CAIC ETIWD TIWD EIWD IWD 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

21 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) [3] G S Mudhoa ad A D Huso Expoad Wu famy: Som pops ad food daa appcao Commu Sasca Thoy ad Mhod 5 (996) [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) [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) [5] R C Gupa ad R D Gupa Popooa vsd hazad mod ad s appcaos J Sa Pa If 37() (7)

22 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