( x) On the Exponentiated Generalized Weibull Distribution: A Generalization of the Weibull Distribution. 1. Introduction.

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1 Indin Journl o Scinc nd Tchnology, Vol 8(35), DOI:.7485/ist/25/v8i35/676, Dcmr 25 ISSN (Print) : ISSN (Onlin) : On th Eponntitd Gnrlizd Wiull Distriution: A Gnrliztion o th Wiull Distriution P. E. Oguntund, O. A. Odtunmii nd A. O. Adumo 2 Dprtmnt o Mthmtics, Covnnt Univrsity, Ot, Ogun Stt, Nigri; plummn@yhoo.com, oluwol.odtunmii@covnntunivrsity.du.ng 2 Dprtmnt o Sttistics, Univrsity o Ilorin, Ilorin, Kwr Stt, Nigri; odumo@gmil.com Astrct Bcground/Octivs: In this rticl, gnrliztion o th Wiull distriution is ing studid in som dtils. Th nw modl is rrrd to s th Eponntitd Gnrlizd Wiull distriution. Th im is to incrs th liility o th Wiull distriution. Mthods: Th concpts introducd in th Eponntitd Gnrlizd mily o distriutions du to Cordiro t l. wr mployd. Findings: Som sic mthmticl proprtis o th rsulting modl wr idntiid nd studid in minut dtils. Mnwhil, stimtion o modl prmtrs ws prormd using th mimum lilihood mthod. Appliction/Improvmnt: Th Eponntitd Gnrlizd Wiull distriution ws prsntd s comptitiv modl tht would usul in modling rl li situtions with invrtd thtu ilur rts. Th R-cod or th plots ws lso providd. Furthr rsrch would involv pplying th proposd modl to rl li dt sts. Kywords: Eponntitd Gnrlizd Wiull Distriution, Gnrliztion, Invrtd Bthtu Filur Rts, Wiull Distriution. Introduction In proility distriution thory, th Wiull distriution is widly nown s ing vrstil, rltivly simpl nd it hs rcivd pprcil usg in th ilds o nginring, mdicl scincs, wthr orcsting, insurnc nd mny mor in rcnt tims. It ws idntiid y nd ws nmd tr 2. In id to incrs th liility o stndrd thorticl proility modls, svrl notl uthors hv proposd vrious gnrliztions or gnrlizd modls; S 3 5. Attmpts to gnrliz th Wiull distriution includ th wor o 6 who introducd th Gnrlizd Wiull (GW) distriution with thtu hzrd (or ilur) rt nd th wor o 7 who dind th Bt Wiull distriution. Mny othrs r vill in th litrtur. According to 8, th wors o 9 nd dmonstrtd th potntility o th GW distriution in nlyzing dt sts rlting to us motor ilur, hd nd nc cncr, nd lood. Th Proility Dnsity Function (pd) nd th Cumultiv Dnsity Function (cd) o th GW distriution r givn y; ( ) = l l ( l ) For >, >, >, l > ( l ) nd F ( ) For >, >, >, l > rspctivly. Whr, α nd β r shp prmtrs. λ is scl prmtr () (2) *Author or corrspondnc

2 On th Eponntitd Gnrlizd Wiull Distriution: A Gnrliztion o th Wiull Distriution On th contrry, this rticl intnds to tnd th Wiull distriution using th Eponntitd Gnrlizd mily o distriution du to. In othr words, w s to tnd th wors o 7, with viw to dining nd invstigting our prmtr Eponntitd Gnrlizd Wiull distriution in th sm wy th Invrs Eponntil distriution ws gnrlizd in 2. According to, th pd nd cd o th Eponntitd Gnrlizd (EG) clss o distriutions r givn y; g( ) G G (3) nd th corrsponding cumultiv distriution unction is givn y; F G = (4) rspctivly. Whr,, > r dditionl shp prmtrs. G() is th cd o th slin (or prnt) distriution dg( ) nd g = d Eqution (4) is sir to hndl nd rltivly simplr thn th Bt-G mily o distriutions du to 3, this is du to th ct tht, Eqution (4) dos not includ spcil unctions li th incomplt t unction. Also, Eqution (4) hs trctility dvntg or simultion purposs cus its quntil unction ts simpl orm,2. For dtild inormtion on th physicl intrprttion o Eqution (4), rdrs r rrrd to. Th rst o this rticl is structurd s ollows; in Sction 2, th Eponntitd Gnrlizd Wiull (EGWiull) distriution is dind, in Sction 3, prssions or som sic sttisticl proprtis o th modl r providd, procdur or stimting th modl prmtrs is givn in Sction 4, ollowd y concluding rmr. Th R-cod usd or th plots in this rsrch is providd s APPENDIX. 2. Th Eponntitd Gnrlizd Wiull Distriution Th pd nd cd o th Wiull distriution with prmtrs α nd β r givn y; g( )= nd = ( ) ; >, >, > (5) G ; >, >, > (6) rspctivly. Whr, α is th shp prmtr. β is th scl prmtr. Also, th mn is givn y; nd; E X = Γ + (7) Mdin = ln 2 (8) 2 Vr X = Γ 2 Γ Following th contnts o,2, th Eponntitd Gnrlizd Wiull (EGWiull) distriution is drivd y sustituting Equtions (5) nd (6) into Eqution (3). In mor clr trm, i rndom vril X is such tht; X EGWiull(,,, ), thn its pd is givn y; (9) ( )= ( ) ( ) () ( ) For >, >, >, >, > Th prssion in Eqution () cn rducd to; ( )= Thror; ( )= For ( ) ( ) >, >, >, >, > ( ) Whr, nd α r shp prmtrs. β is th scl prmtr. () 2 Vol 8 (35) Dcmr 25 Indin Journl o Scinc nd Tchnology

3 P. E. Oguntund, O. A. Odtunmii nd A. O. Adumo Th ssocitd cd o th EGWiull distriution is givn y; F( )= >, >, >, >, > (2) Th prssion in Eqution (2) is simpliid to giv; F = ( ) >, >, >, >, > 2. Epnsions or th CDF (3) For ny rl non-intgr, in, considrd powr sris pnsion givn y; ( z) = ( ) Γ( ) Γ! z (4) It is good to not tht th prssion in Eqution (4) is vlid or z <. Using th inomil pnsion or positiv rl powr, th rsulting cd is givn y; F wg( ) Th coicints w w, wr givn y; w = + ( ) ( + ) + = Γ Γ Γ!! (5) With this nowldg, th cd o th EGWiull distriution cn r-writtn s; = F + ( ) Γ( + ) Γ( + ) Γ ( )!! (6) t ( = ) Γ Γ! Γ Γ! ( + ) ( + ) (8) Thror, w r-writ th pd o th EGWiull distriution s; = ( ) ( ) Γ( )! ( ) Γ ( + ) ( ) Γ( ) Γ +! 2.2 Rltionship with Othr Distriutions (9) Som importnt modls in th litrtur r spcil css o th EGWiull distriution. For mpl, For =, Eqution () rducs to giv th Gnrlizd Wiull (GW) distriution. For =, Eqution () rducs to giv th EponntitdWiull (EW) distriution. For ==, Eqution () rducs to giv th Wiull distriution (which is th slin distriution). For == α =, Eqution () rducs to giv th Eponntil distriution. For rvity, plot or th pd o th EGWiull distriution t =2, =2, α =2, β =3, is givn in Figur ; Th plot s shown in Figur indicts tht th shp o th EGWiull distriution could unimodl. This would urthr conirmd in th nt sction. A possil plot or th cd o th EGWiull distriution t =2, =2, α =2, β =3, is givn in Figur 2; It is good to not tht th prssion in Eqution (6) is n ininit powr sris o th Wiull distriution. With rspct to th sris pnsion in Eqution (4), uthor in gv th pd o th Eponntitd Gnrlizd (EG) clss o distriutions (or rl non-intgr) s; = g tg Th coicints t t, wr = (7) Figur. Plot or th pd o th EGWiull distriution =2, =2, α =2, β =3. Vol 8 (35) Dcmr 25 Indin Journl o Scinc nd Tchnology 3

4 On th Eponntitd Gnrlizd Wiull Distriution: A Gnrliztion o th Wiull Distriution = As Æ ; lim F = lim = Figur 2. Plot or th cd o th EGWiull distriution =2, =2, α =2, β =3. 3. Proprtis o th EGWiull Distriution Som sic sttisticl proprtis o th EGWiull distriution r idntiid in this sction s ollows 3. Limiting Bhvior Th hvior o th pd o EGWiull distriution is ing invstigtd s Æ nd s Æ. Tht is, lim ( ) nd lim ( ) r considrd As Æ; lim lim = ( ) ( ) ( ) Not: This is cus; s Æ, th prssion coms zro. As Æ ; lim lim = ( ) ( ) ( ) Ths rsults irm tht th proposd EGWiull distriution hs uniqu mod. In th sm wy, th hvior o th cd o EGWiull distriution s givn in Eqution (3) is ing invstigtd s Æ nd s Æ. Tht is, lim F nd lim F r considrd. As Æ; lim F = lim 3.2 Rliility Anlysis Hr, prssions or th survivor (or rliility) unction nd th hzrd rt o th EGWiull distriution r providd. Mthmticlly, th rliility unction is givn y; S( ) P { X> } u du F = = Thror, th survivor unction o th EGWiull distriution is prssd s; SEGWiull ( ) (2) For >, >, >, >, > A plot or th survivor unction o th EGWiull distriution is givn in Figur 3; Mthmticlly, hzrd (or ilur) rt is givn y; = h F( ) Thror th hzrd rt or th EGWiull distriution is givn y; Figur 3. Plot or th survivor unction o th EGWiull distriution =2, =2, α =2, β =3. 4 Vol 8 (35) Dcmr 25 Indin Journl o Scinc nd Tchnology

5 P. E. Oguntund, O. A. Odtunmii nd A. O. Adumo = h ( ) ( ) (2) For >, >, >, >, > A plot or th hzrd unction t =2, =2, α =2, β =3 is s shown in Figur 4; Th plot in Figur 4 shows tht th EGWiull distriution hs n invrtd thtu ilur rt. This implis tht th EGWiull distriution cn usd to modl rl li situtions with invrtd thtu ilur rts. 3.3 Momnts According to, th momnts o ny EG distriution cn prssd s n ininit wightd sum o proility wightd momnts o th prnt distriution. E r X = = t t τ r in this cs is sd on th quntil unction o th Wiull distriution; Q u G Following, lt G u ; r G = r t r = G Q u udu (22) t is s dind in Eqution (8). In prticulr, th quntil unction or th Wiull distriution is givn y; Figur 4. Plot or th hzrd rt o th EGWiull distriution =2, =2, α =2, β =3.. Thror; QG ( u) = ln u (23) r r tr = ( ) u ln ( u) du (24) Insrting Equtions (8) nd (24) into Eqution (22) givs th prssion or th rth momnt o th EGWiull distriution. 3.4 Quntil Function Th quntil unction Q hs n idntiid s n ltrntiv to th pd s it is wy o prscriing proility distriution. It is drivd s th invrs o th cd. With this undrstnding, th quntil unction o th EGWiull distriution cn prssd s; Qu u = ln (25) Sustituting u =.5 givs th mdin. Thror, th mdin is givn y; Q 5. = ln (. 5) (26) Rndom vrils rom th EGWiulldistriution cn gottn using th prssion; 3.5 Ordr Sttistics X u = ln r (27) Th pd o th ith ordr sttistic or,2...,n rom indpndntly nd idnticlly distriutd rndom vrils X, X,..., X is givn y; 2 2 i F( ) F( ) Bin, i+ (28) Following, th distriution o ordr sttistics or th EG mily o distriutions is prssd s; i g( ) G G Bin, i { } G (29) Vol 8 (35) Dcmr 25 Indin Journl o Scinc nd Tchnology 5

6 On th Eponntitd Gnrlizd Wiull Distriution: A Gnrliztion o th Wiull Distriution With th id o inomil pnsion, th distriution o ordr sttistics or th EGWiull distriution cn writtn s; Bin, i ( ) n i ( ) ( ) i ( + ) (3) Th prssion in Eqution (3) cn simpliid to giv; Bin, i ( ) 4. Estimtion ( ) ( ) ( + ) i (3) n i Th prmtrs o th EGWiull distriution cn stimtd using th mthod o mimum lilihood. Lt X, X 2,...,X n dnot rndom smpl o siz n rom th EGWiull (,, α, β) distriution. Th lilihood unction is givn y; n L X,,, = ( ) Din l= log L X,,, Thror, th log-lilihood unction is givn y; l= nlog+ nlog+ nlog n log + n i log n n i + log (32) i (33) l Th solutions o = l = l = l nd = givs th mimum lilihood stimt or prmtrs,, α, nd β rspctivly. Th solution o th non-linr systms o qutions my not sily drivd nlyticlly ut cn gottn numriclly using sttisticl sotwr li SAS or R. 5. Conclusion This rticl studis our prmtr Eponntitd Gnrlizd Wiull distriution y gnrlizing th two-prmtr wiull distriution. Th im is to induc swnss into th prnt distriution in ordr to incrs its liility nd cpility to modl dt sts tht r mor swd. Th modl is unimodl nd it hs th Gnrlizd Wiull distriution, Eponntitd Wiull distriution, Wiull distriution nd th Eponntil distriution s su modls. Som sic proprtis o th EGWiull distriution r idntiid nd th modl is cpl o modling rl li situtions with invrtd thtu shp. Furthr rsrch would involv n ppliction o th EGWiull distriution to rl dt st to ssss its liility ovr its su modls. APPENDIX R-cod or th plots; > =sq(,2,.) > [] [24] [47] [7] [93] [6] [39] [62] [85] > =2 > =2 > c=2 > d=3 > pd=**(c/d)*(/d)^(c-)*(p(-(/d)^c))^*((- p(-(/d)^c))^)^(-) >plot(,pd,min= PDF o th EGWiull Distriution ) > =2 > =2 6 Vol 8 (35) Dcmr 25 Indin Journl o Scinc nd Tchnology

7 P. E. Oguntund, O. A. Odtunmii nd A. O. Adumo > c=2 > d=3 >cd=(-(p(-(/d)^c))^)^ >plot(,cd,min= CDF o th EGWiull Distriution ) >SurvivlFunction=-cd >plot(,survivlfunction,min= Survivl Function o th EGWiull Distriution ) >HzrdRt=pd/SurvivlFunction >plot(,hzrdrt,min= Hzrd Rt o th EGWiull Distriution ) 6. Rrncs. Frcht M. Sur l loi d proilit d l crt mimum. Annls d l Socit Polonis d Mthmtiqu, Crcovi. 927; 6: Wiull W. A sttisticl distriution unction o wid pplicility. J Appl Mch - Trns ASME. 95; 8(3): Alshwrh E. Bt-cuchy distriution nd its pplictions [Ph D thsis]. Michign, USA: Cntrl Michign Univrsity; Bourguignon M, Silv RB, Cordiro GM. Th Wiull-G mily o proility distriutions. Journl o Dt Scinc. 24; 2: An TA, Oguntund PE, Odtunmii OA. On rctionl t-ponntil distriution. Intrntionl Journl o Mthmtics nd Computtion. 25; 26(): Mudholr GS, Srivstv DK. Eponntitd Wiull mily or nlyzing thtu ilur-rt dt. IEEE Trnsctionson Rliility. 993; 42(2): L C, Fmoy F, Olumold O. Bt-Wiull distriution: Som proprtis nd pplictions to cnsord dt. Journl o Modrn Applid Sttisticl Mthods. 27; 6(). Articl Jin K, Singl N, Shrm SK. Th gnrlizd invrs gnrlizd Wiull distriution nd its proprtis. Journl o Proility. 24:. Articl ID 736. Avill rom: 9. Mudholr GS, Srivstv DK, Frimr M. Th ponntitd wiull mily: A rnlysis o th us-motor ilur dt. Tchnomtrics. 995: 37(4): Mudholr GS, Hutson AD. Th ponntitd Wiull mily: Som proprtis nd lood dt ppliction. Communictionsin Sttistics: Thory nd Mthods. 996; 25(2): Cordiro GM, Ortg EM, d Cunh DC. Th ponntitd gnrlizd clss o distriutions. Journl o Dt Scinc. 23; : Oguntund PE, Adumo AO, Blogun OS. Sttisticl proprtis o th ponntitd gnrlizd invrtd ponntil distriution. Applid Mthmtics. 24; 4(2): Eugn N, L C, Fmoy F. Bt-norml distriution nd its pplictions. Communiction in Sttistics: Thory nd Mthods. 22; 3(4): Vol 8 (35) Dcmr 25 Indin Journl o Scinc nd Tchnology 7