Odd Generalized Exponential Flexible Weibull Extension Distribution

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1 Odd Gralzd Epotal Flbl Wbull Etso Dstrbuto Abdlfattah Mustafa Mathmatcs Dpartmt Faculty of Scc Masoura Uvrsty Masoura Egypt abdlfatah Bh S. El-Dsouy Mathmatcs Dpartmt Faculty of Scc Masoura Uvrsty Masoura Egypt b dsouy@yahoo.com Shamsa AL-Garash Mathmatcs Dpartmt Faculty of Scc Masoura Uvrsty Masoura Egypt shamsa algarash@hotmal.com Rcvd 1 Dcmbr 16 Accptd 9 Dcmbr 17 I ths artcl w troduc a w four - paramtrs modl calld th odd gralzd potal flbl Wbull tso OGE-FWE dstrbuto whch hbts bathtub-shapd hazard rat. Som of t s statstcal proprts ar obtad cludg ordary ad complt momts quatl ad mod th momt gratg fuctos rlablty ad ordr statstcs. Th mthod of mamum llhood s usd for stmatg th modl paramtrs ad th obsrvd Fshr s formato matr s gv. Morovr w gv th advatag of th OGE-FWE dstrbuto by a applcato usg ral data. Kywords: Odd Gralzd Epotal Famly; Flbl Wbull Etso dstrbuto; Gralzd Wbull; Odd Gralzd Epotal Flbl Wbull dstrbuto; Mamum llhood stmato. Mathmatcs Subjct Classfcato: 9B5 6N5 6P3 1. Itroducto Th Wbull dstrbuto s a hghly ow dstrbuto du to ts utlty modllg lftm data whr th hazard rat fucto s mooto 7. Rctly w classs of dstrbutos wr proposd basd o modfcatos of th Wbull dstrbuto to provd a good ft to data st wth bathtub hazard falur rat s 5. Amog of ths Epotatd Wbull famly 14 Modfd Wbull dstrbuto Bta-Wbull dstrbuto 8 A flbl Wbull tso 4 Etdd flbl Wbull 4 Gralzd modfd Wbull dstrbuto 5 Kumaraswamy Wbull dstrbuto 6 Bta modfd Wbull dstrbuto 16 Bta gralzd Wbull dstrbuto 3 Copyrght 18 th Authors. Publshd by Atlats Prss. Ths s a op accss artcl udr th CC BY-NC lcs 77

2 A w modfd wbull dstrbuto ad Epotatd modfd Wbull tso dstrbuto amog othrs. A good rvw of ths modls s prstd Th flbl Wbull tso FWE dstrbuto 4 has may applcatos lf tstg prmts rlablty aalyss appld statstcs ad clcal studs. For mor dtals o ths dstrbuto s 4. A radom varabl X s sad to hav th Flbl Wbull Etso FWE dstrbuto wth paramtrs α > f t s probablty dsty fucto pdf s gv by { } α g = α + p α > 1.1 whl th cumulatv dstrbuto fucto cdf s gv by { } G = 1 p α >. 1. Th survval fucto s gv by th quato { } S = 1 G = p α > ad th hazard rat fucto s h = α + α Gupta ad Kudu 9 proposd a gralzato of th potal dstrbuto amd as Gralzd Epotal GE dstrbuto. Th GE dstrbuto wth paramtrs γ > has th followg dstrbuto fucto γ F; γ = 1 > > γ >. 1.5 Rctly a w class of uvarat cotuous dstrbutos amd as th odd gralzd potal OGE class troducd 7 4. Ths class s flbl bcaus of th hazard rat shaps could b crasg dcrasg bathtub ad upsd dow bathtub. Th odd gralzd potal OGE class s dfd as follows. If G > s cumulatv dstrbuto fucto cdf of a radom varabl X th th corrspodg survval fucto s G = 1 G ad th probablty dsty fucto s g th w df th cdf of th OGE class by rplacg th dstrbuto fucto of gralzd potal GE G ladg to dstrbuto gv quato 1.5 by G { } G γ F; γ = 1 p > > γ >. G Th probablty dsty fucto corrspodg to 1.6 s gv by { } { } γ g G G γ f ; γ = p 1 p G G G whr > > γ >. I ths artcl w prst a w dstrbuto dpdg o flbl Wbull tso dstrbuto rfrrd to as th odd gralzd potal flbl Wbull tso 78

3 OGE-FWE dstrbuto by usg th class of uvarat dstrbutos dfd abov. Ths papr ca b orgazd as follows w df th cumulatv dsty ad hazard fuctos of th odd gralzd potal flbl Wbull tso OGE-FWE dstrbuto Scto. I Sctos 3 w prst som statstcal proprts cludg quatl fucto ad mda th mod rth momt swss ad urtoss. I Sctos 4 w troduc th momt gratg fucto. Th dstrbuto of th ordr statstcs s prssd Scto 5. Th mamum llhood stmato of th paramtrs s dtrmd Scto 6. W us ral data sts ad aalyzd t by a applcato Scto 7 ad th rsults ar compard wth stg dstrbutos. Fally w prst a cocluso Scto 8.. Th Odd Gralzd Epotal Flbl Wbull Etso Dstrbuto I ths scto w studd th four paramtrs odd gralzd potal flbl Wbull tso OGE-FWE γ α dstrbuto. Usg G from Eq. 1. ad g from Eq. 1.1 to obta th cdf ad pdf of Eqs. 1.6 ad 1.7 rspctvly. Th cumulatv dstrbuto fucto cdf of th odd gralzd potal flbl Wbull tso dstrbuto OGE-FWE s gv by α γ F; γ α = 1 > γ α >.1 Th pdf corrspodg to Eq..1 s gv by γ α α +α α f ; γ α = γ α + 1. whr > ad α > ar two addtoal shap paramtrs. Th survval fucto S hazard rat fucto h ad rvrsd hazard rat fucto r of X OGE-FWE γ α ar gv by γ α S; γ α = 1 1 >.3 γ α α α +α γ α + 1 h; γ α = γ α 1 1 r; γ α = α α +α γ α + α rspctvly > ad γ α >. Fgurs 1-3 dsplay th cdf pdf survval hazard rat ad rvrsd hazard rat fucto of th OGE-FWE γ α dstrbuto for som paramtr valus. 79

4 a cdf b pdf Fg. 1. Th cdf ad pdf of th OGE-FWE for dffrt valus of paramtrs. a S b h Fg.. Th survval ad hazard rat fuctos of th OGE-FWE for dffrt valus of paramtrs. Fg. 3. Th rvrsd hazard rat fucto of th OGE-FWE for dffrt valus of paramtrs. 3. Statstcal Proprts I ths scto w wll study som statstcal proprts for th OGE-FWE dstrbuto spcally quatl fucto ad smulato mda th mod momts swss ad urtoss Quatl ad mda W dtrm th plct formulas of th quatl ad smulato mda of th OGE-FWE dstrbuto. Th quatl q of th OGE-FWE γ α dstrbuto s gv by Fq = Pq q = q < q <

5 From Eq..1 w hav 1 α q q γ = q 3. w obta q by solvg th followg quato. α q qq = whr { 1 l1 q γ q = l l }. So th smulato of th OGE-FWE radom varabl s straghtforward. If U s a uform radom varabl o ut trval 1. Th by mas of th vrs trasformato mthod w ca obta th radom varabl X as follows u ± u + 4α X=. 3.4 α Sc th mda of OGE-FWE dstrbuto ca b obta by sttg q =.5 Eq Th mod I ths subscto w ca obta th mod of th OGE-FWE dstrbuto by dffrtatg ts probablty dsty fucto pdf wth rspct to ad qualg t to zro. Th mod s th soluto th followg quato f =. 3.5 Sc f ; γ α = h; γ α S; γ α. Th from Eq. 3.5 w hav h ; γ α h ; γ α S; γ α = 3.6 whr h; γ α s hazard rat fucto of OGE-FWE dstrbuto Eq..4 ad S; γ α s survval fucto of OGE-FWE Eq..3. It s dffcult to gt a aalytc soluto to Eq. 3.6 th gral cas. So t has to b obtad by umrcally mthods Swss ad Kurtoss I ths subscto w obta th swss ad urtoss basd o quatl masurs. Th Moors Kurtoss s gv by 13 Ku = q.875 q.65 q q.15 q.75 q.5 ad th Bowly s swss basd o quartls s gv by

6 q.75 q.5 + q.5 q.75 q.5 S = 3.8 whr q. s quatl fucto Th Momts W drv th rth momt for th OGE-FWE dstrbuto Thorm 3.1 Thorm 3.1. If X has OGE-FWE γ α dstrbuto th Th rth momts of radom varabl X s gv by + j++m γ j j j + 1ℓ m j!ℓ!m!ℓ + 1rm α rm = j= = ℓ= m= γ j Γr m Γr m 1. α ℓ + 1 µr = 3.9 Proof. W start wth th wll ow dstrbuto of th rth momt of th radom varabl X wth probablty dsty fucto f gv by µr = r f ; γ α d. 3.1 Substtutg from Eq.. to Eq. 3.1 w gt γ α α r α α d µr = γ α + 1 γ α α sc < 1 < 1 for > th bomal srs paso of 1 ylds γ α α γ 1 = = th w gt µr α α γ r = γ α + α +1 d = usg srs paso α +1 j j + 1 j = j! j= j α w obta µr = = j= + j j γ 1 γ j j r α α α α+ 1 d. j! 8

7 Usg srs paso j α = j jα = hc µr α γ j + j+ γ j j r α + α j+1 d = j! = j= = usg srs paso j + 1ℓ ℓα = ℓ! ℓ= j+1α w obta µr γ j + j+ γ j j j + 1ℓ r α + ℓ+1α d. = j!ℓ! = j= = ℓ= Usg srs paso ℓ+1 = m ℓ + 1m m m! m= gvs µr + j++m γ j j j + 1ℓ m ℓ + 1m γ 1 j = j!ℓ!m! = j= = ℓ= m= rm α + ℓ+1α d + j++m γ j j j + 1ℓ m ℓ + 1m γ 1 j = j!ℓ!m! = j= = ℓ= m= α rm ℓ+1α d + rm ℓ+1α d by usg th dfto of gamma fucto th form 6 Γz = z tt z dt z >. Fally w obta th rth momt of OGE-FWE dstrbuto th form µr + j++m γ j j j + 1ℓ m ℓ + 1m γ 1 j = j!ℓ!m! = j= = ℓ= m= Γr m + 1 Γr m 1 +. α rm ℓ + 1rm+1 α rm ℓ + 1rm Ths complts th proof. 83

8 4. Th Momt Gratg Fucto Th momt gratg fucto mgf of th OGE-FWE dstrbuto s gv by Thorm 4.1. Thorm 4.1. Th momt gratg fucto mgf of OGE-FWE dstrbuto s gv by MX t = + j++m γ j j j + 1ℓ mt r j!ℓ!m!r!α rm ℓ + 1rm r= = j= = ℓ= m= γ j Γr m Γr m 1. α ℓ Proof. Th momt gratg fucto MX t of th radom varabl X wth probablty dsty fucto f s gv by MX t = t f ; γ α d 4. Usg srs paso of t w obta MX t = tr r= r! r f d = tr µr r= r! 4.3 Substtutg from 3.9 to 4.3 w obta MX t = + j++m γ j j j + 1ℓ mt r j!ℓ!m!r!α rm ℓ + 1rm r= = j= = ℓ= m= γ j Γr m Γr m 1. α ℓ Ths complts th proof. 5. Ordr Statstcs I ths scto w drv closd form prssos for th PDFs of th rth ordr statstc of th OGEFWE dstrbuto. Lt X1: X: X: dot th ordr statstcs obtad from a radom sampl X1 X X whch ta from a cotuous populato wth cumulatv dstrbuto fucto cdf F; φ ad probablty dsty fucto pdf f ; φ th th pdf of Xr: s gv as follows fr: ; φ = 1 F; φ r 1 F; φ r f ; φ Br r whr f ; φ ad F; φ ar th pdf ad cdf of OGE-FWEφ dstrbuto gv by Eq.. ad Eq. 1.7 rspctvly φ = γ α ad B.. s th Bta fucto also w df frst ordr statstcs X1: = mx1 X X ad th last ordr statstcs as X: = max1 X X. Sc 84

9 < F; φ < 1 for > w ca us th bomal paso of 1 F; φ r as follows 1 F; φ r r = = r F; φ. 5. Substtutg from Eq. 5. to Eq. 5.1 w obta fr: ; γ α = r!!r 1! r! f ; φ F; φ +r. 5.3 = Substtutg from Eq..1 ad Eq.. to Eq. 5.3 w obta th probablty dsty fucto for rth ordr statstcs. Rlato 5.3 show that fr: ; φ s th wghtd avrag of th OGE-FWE dstrbuto wth dffrt shap paramtrs. 6. Paramtrs Estmato I ths scto pot ad trval stmato of th uow paramtrs of th OGE-FWE dstrbuto ar drvd by usg th mamum llhood mthod basd o a complt sampl Mamum Llhood Estmato: Lt 1 dot a radom sampl of complt data from th OGE-FWE dstrbuto. Th Llhood fucto s gv as L = f ; γ α 6.1 =1 substtutg from. to 6.1 w hav γ α α α α L = γ α + 1. =1 Th log-llhood fucto s α α L = l γ + l α + + α + =1 =1 =1 =1 α +γ 1 l =1 Th mamum llhood stmato of th paramtrs γ α ar obtad by dffrtatd th log-llhood fucto L wth rspct to th paramtrs γ α ad ad sttg th rsult to zro w hav th followg ormal quatos. 85

10 α L = 1 + γ 1 =1 =1 α L = + l 1 = γ γ =1 α α = L α = + + D + γ 1 α =1 + α =1 =1 =1 =1 1 L 1 1 = =1 + α =1 =1 α D α = D D γ 1 = α =1 = } { α whr D = p α +. Th MLEs ca b obtad by solvg th quatos umrcally for γ α ad. 6.. Asymptotc cofdc bouds I ths scto w drv th asymptotc cofdc trvals of ths paramtrs wh γ α > ad > as th MLEs of th uow paramtrs γ α > ad > ca ot b obtad closd forms by usg varac covarac matr I s 1 whr I s th vrs of th obsrvd formato matr whch dfd as follows I L L L L γ a α L γ γl γ Lα γ L = L L L L α α γ α α L L L L γ α var cov γ cov α cov covγ varγ covγ α covγ = covα covα γ varα covα cov cov γ cov α var 6.7 whr α L = γ 1 A =1 L = D + γ 1 B α =1 =1 L = A γ =1 6.8 L D B γ 1 = =1 =1 L = γ γ L = γ α =1 86 D α

11 D L = α γ = L 4 α α γ 1 = + D + α K6.1 α =1 =1 =1 =1 L α α + = D γ K α =1 + α =1 =1 =1 α 1 L = + =1 + α =1 α D + 1 =1 K = γ whr α α A = α α α 1 1 B = D 1 α α α α + 1 D K = D 1. W ca obta th 1 δ 1% cofdc trvals of th paramtrs γ α ad by usg varac matr as th followg forms ± Z δ var γ ± Z δ varγ α ± Z δ varα ± Z δ var whr Z δ s th uppr δ -th prctl of th stadard ormal dstrbuto. 7. Applcato I ths scto w wll aalyss of a ral data st usg th OGE-FWE γ α modl ad compar t wth th othr fttd modls l a flbl Wbull tso dstrbutos usg Kolmogorov Smrov K-S statstc as wll as Aa formato crtroaic? Aa Iformato Ctro wth corrcto AICC Baysa formato crtro BIC Haa-Qu formato crtro HQIC ad Schwarz formato crtro SIC valus 1. Th data hav b obtad from 18 t s for th tm btw falurs thousads of hours of scodary ractor pumps Tabl 1. Tabl 1. Tm btw falurs of scodary ractor pumps

12 Tabl. MLEs ad KS of paramtrs for scodary ractor pumps. Modl OGE-FWE Flbl Wbull Wbull Modfd Wbull Rducd Addtv Wbull Etdd Wbull α λ γ.113 K-S Tabl 3. Log-llhood AIC AICC BIC HQIC ad SIC valus of modls fttd. Modl OGE-FEW Flbl Wbull Wbull Modfd Wbull Rducd addtv Wbull Etdd Wbull L AIC AICC BIC HQIC SIC Tabl gvs MLEs of paramtrs of th OGE-FWE ad K-S Statstcs. Th valus of th logllhood fuctos AIC AICC BIC HQIC ad SIC ar Tabl 3. Substtutg th MLEs of th uow paramtrs γ α to 6.7 w obta stmato of th varac covarac matr as th followg I = Th appromat 95% two sdd cofdc trvals of th uow paramtrs γ α ad ar ad rspctvly. Th oparamtrc stmat of th survval fucto S usg th Kapla-Mr mthod ad ts fttd paramtrc stmatos wh th dstrbuto s assumd to b OGE-FWE FW W MW RAW ad EW ar computd ad plottd Fgur 4. Fg. 4. Th Kapla-Mr stmat of th survval fucto for th data. 88

13 Fgur 5 gvs th form of th hazard rat h ad cumulatv dsty fucto cdf for th OGEFWE FW W MW RAW ad EW whch ar usd to ft th data aftr th uow paramtrs cludd ach dstrbuto ar rplacd by thr MLEs. a ht b cdf Fg. 5. Th Fttd hazard rat ad cumulatv dstrbuto fucto for th data. W fd that th OGE-FWE dstrbuto wth th four - paramtrs provds a bttr ft tha th prvous w modfd a flbl Wbull tso dstrbutofwe whch was th bst 4. It has th largst llhood ad th smallst AIC AICC BIC HQIC ad SIC valus amog thos cosdrd ths papr. 8. Coclusos W proposd a w dstrbuto basd o odd gralzd potal famly dstrbutos ths dstrbuto s amd th odd gralzd potal flbl Wbull tso OGE-FWE dstrbuto. Som ts statstcal proprts ar studd. Th mamum llhood mthod s usd for stmatg th paramtrs modl. Fally w troduc a applcato usg ral data. W show that th OGE-FWE dstrbuto fts crta wll ow data sts bttr tha stg modfcatos of th Wbull ad flbl Wbull tso dstrbutos. Rfrcs 1 H. Aa A w loo at th statstcal modl dtfcato IEEE Trasactos o Automatc Cotrol AC S.J. Almal ad J. Yua Th w modfd Wbull dstrbuto Rlablty Egrg ad Systm Safty E. K. AL-Hussa ad M. Ahsaullah Epotatd Dstrbutos. Atlats-Prss Pars Frac M.S. Bbbgto C.D. La ad R. Zts A flbl Wbull tso. Rlablty Egrg & Systm Safty M. Carrasco E.M. Ortga ad G.M. Cordro A gralzd modfd Wbull dstrbuto for lftm modlg Computatoal Statstcs ad Data Aalyss G.M. Cordro E.M. Ortga ad S. Nadarajah Th Kumaraswamy Wbull dstrbuto wth applcato to falur data. Joural of th Fral Isttut M. A. El-Damcs A. Mustafa B. S. El-Dsouy ad M. E. Mustafa Th odd gralzd potal gomprtz arxv prprt arxv: F. Famoy C. L ad O. Olumolad Th bta-wbull dstrbuto Joural of Statstcal Thory ad Applcatos

14 9 R.D. Gupta ad D. Kudu Gralzd potal dstrbuto: Estg rsults ad som rct dvlopmts Joural of Statstcal Plag ad Ifrc J. Ky ad E. Kpg Mathmatcs of Statstcs Prcto C.D. La M. X ad D.N.P. Murthy A modfd Wbull dstrbutos IEEE Trasactos o Rlablty J. F. Lawlss Statstcal Modls ad Mthods for Lftm Data Joh Wly ad Sos Nw Yor J.J.A. Moors A quatl altratv for urtoss Th Statstca G.S. Mudholar ad D.K. Srvastava Epotatd Wbull famly for aalyzg bathtub falur-rat data IEEE Trasactos o Rlablty D. N. P. Murthy M. X ad R. Jag Wbull Modls Joh Wly ad Sos Nw Yor S. Nadarajah G.M. Cordro ad E.M.M. Ortga Gral rsults for th bta-modfd Wbull dstrbuto Joural of Statstcal Computato ad Smulato H. Pham ad C.D. La O rct gralzatos of th Wbull dstrbuto IEEE Trasactos o Rlablty M. Salma Suprawhardaa ad S. Prayoto Total tm o tst plot aalyss for mchacal compots of th RSG-GAS ractor Atom Idos A.M. Sarha ad M. Zad Modfd Wbull dstrbuto Appld Sccs A.M. Sarha ad J. Apaloo Epotatd modfd Wbull tso dstrbuto Rlablty Egrg ad Systm Safty G. Schwarz Estmatg th dmso of a modl Aals of Statstcs G.O. Slva E.M. Ortga ad G.M. Cordro Th bta modfd Wbull dstrbuto Lftm Data Aalyss N. Sgla K. Ja ad S. S. Kumar Th bta gralzd Wbull dstrbuto: proprts ad applcatos Rlablty Egrg & Systm Safty M. H. Tahr G. M. Cordro M. Alzadh M. Masoor M. Zubar ad G. G. Hamda Th odd gralzd potal famly of dstrbutos wth applcatos Joural of Statstcal Dstrbutos ad Applcatos M. X ad C.D. La Rlablty aalyss usg a addtv Wbull modl wth bathtub-shapd falur ratfucto Rlablty Egrg Systm Safty D. Zwllgr Tabl of tgrals srs ad products Elsvr W.A. Wbull Statstcal dstrbuto fucto of wd applcablty Joural of Appld Mchacs