Classes of Ordinary Differential Equations Obtained for the Probability Functions of 3-Parameter Weibull Distribution

Transcription

1 Prodings of h World Congrss on Enginring and Compur Sin 07 Vol II WCECS 07, Oor 5-7, 07, San Franiso, USA Classs of Ordinary Diffrnial Equaions Oaind for h Proailiy Funions of 3-Paramr Wiull Disriuion Hilary I. Okagu, Mmr, IAENG, Muminu O. Adamu, Aiodun A. Opanuga, Jimvwo G. Oghonyon Asra In his papr, h diffrnial alulus was usd o oain som lasss of ordinary diffrnial quaions (ODE) for h proailiy dnsiy funion, quanil funion, survival funion, invrs survival funion, hazard funion and rvrsd hazard funion of h 3-paramr Wiull disriuion. Th sad nssary ondiions rquird for h xisn of h ODEs ar onsisn wih h various paramrs ha dfind h disriuion. Soluions of hs ODEs y using numrous availal mhods ar a nw ways of undrsanding h naur of h proailiy funions ha harariz h disriuion. Indx Trms 3-paramr Wiull disriuion, diffrnial alulus, proailiy dnsiy funion, survival funion, quanil funion. T I. INTRODUCTION HE 3-paramr Wiull disriuion is a varian of h Wiull disriuion and was oaind o improv h flxiiliy of modling wih Wiull disriuion []. Th disriuion has n sudid y [], whr hy simad h shap paramr of h disriuion. Cran [3] sudid xnsivly h propris of momn simaors of h disriuion whil [4] proposd h rous simaor for h 3-paramr Wiull disriuion. Som ohr asps ha hav n sudid inluds: ondiional xpaion [5], paramr simaion undr dfind nsoring [6-7], nsoring sampling [8], posrior analysis and rliailiy [9-0], minimum and rous minimum disan simaion [- ], hr-paramr Wiull quaions [3], onfidn limis [4], quanil asd poin sima of h paramrs [5], prnil simaion [6], mhods of simaion of paramrs [7-]. Srong ompuaional hniqus hav now n usd in h simaion of paramrs of h disriuion suh as paril swarm opimizaion [], diffrnial voluion [3]. Li [4] applid h las squar mhod in h simaion of h paramrs of h disriuion. Mahmoud [5] osrvd ha h 3-paramr invrs Gaussian disriuion an usd and apply as an alrnaiv modl for h 3-paramr Wiull disriuion. Th disriuion has n usd in h modling of ral lif Manusrip rivd Jun 30, 07; rvisd July 3, 07. This work was sponsord y Covnan Univrsiy, Oa, Nigria. H. I. Okagu, A. A. Opanuga and J.G. Oghonyon ar wih h Dparmn of Mahmais, Covnan Univrsiy, Oa, Nigria. hilary.okagu@ovnanunivrsiy.du.ng aiodun.opanuga@ovnanunivrsiy.du.ng M. O. Adamu is wih h Dparmn of Mahmais, Univrsiy of Lagos, Akoka, Lagos, Nigria. ISBN: ISSN: (Prin); ISSN: (Onlin) siuaions suh as: faigu rak growh [6], sp-srss alrad lif s [7], aging [8], hliopr lad rliailiy [9], os simaion [30], im wn failurs of mahin ools [3]. Th aim of his rsarh is o dvlop ordinary diffrnial quaions (ODE) for h proailiy dnsiy funion (PDF), Quanil funion (QF), survival funion (SF), invrs survival funion (ISF), hazard funion (HF) and rvrsd hazard funion (RHF) of 3-paramr Wiull disriuion y h us of diffrnial alulus. Calulus is a vry ky ool in h drminaion of mod of a givn proailiy disriuion and in simaion of paramrs of proailiy disriuions, amongs ohr uss. Th rsarh is an xnsion of h ODE o ohr proailiy funions ohr han h PDF. Similar works don whr h PDF of proailiy disriuions was xprssd as ODE whos soluion is h PDF ar availal. Thy inlud: Lapla disriuion [3], a disriuion [33], raisd osin disriuion [34], Lomax disriuion [35], a prim disriuion or invrd a disriuion [36]. II. PROBABILITY DENSITY FUNCTION Th proailiy dnsiy funion of h 3- paramr Wiull disriuion is givn as; x x f( x) () wih h paramrs,,, 0, x 0. To oain h firs ordr ordinary diffrnial quaion for h proailiy dnsiy funion of h 3-paramr Wiull disriuion, diffrnia quaion (), o oain; x x () x x x x x (3) WCECS 07

2 Prodings of h World Congrss on Enginring and Compur Sin 07 Vol II WCECS 07, Oor 5-7, 07, San Franiso, USA x,,, 0 Th diffrnial quaions an only oaind for pariular valus of, and. Whn, quaion (3) oms; f a( x) fa( x) (4) f a( x) fa( x) 0 (5) Whn, quaion (3) oms; ( x ) f ( x) f ( ) x x (6) ( x ) f ( x) ( ( x ) ) f( x) 0 (7) Whn 3, quaion (3) oms; 3( x ) f ( x) f ( ) 3 x x (8) ( x ) ( 3( x ) ) 0 (9) Equaion (3) is diffrniad o oain; x x ( ) x ( x ) f( x) (0) Th following quaions oaind from (3) ar ndd o simplify quaion (0); x x x x () () ( ) x x (3) ( ) x x x Susiu quaions () and (4) ino quaion (0); ( x ) f( x) f( x) x x (4) (5) f ( x) ( ) f( x) ( x ) ( ) ( ) ( x) x ( ) ( ) ( ) ( ) ( ) f( x) f x f x f x ( x ) x (6) (7) x 0, x 0, 0,, 0 Th sond ordr ordinary diffrnial quaion for h proailiy dnsiy funion of h 3-paramr Wiull disriuion is givn y; ( x ) f ( x) ( x ) f ( x) ( ) ( )( x ) 0 f (0) (8) (9) f (0) III. QUANTILE FUNCTION (0) Th Quanil funion of h 3- paramr Wiull disriuion is givn as; Q( p) ( ln( p)) () Th paramrs ar:, 0,0 p. To oain h firs ordr ordinary diffrnial quaion for h Quanil funion of h 3-paramr Wiull disriuion, diffrnia quaion (), o oain; Q( p) ( ln( p)) ( p) (), 0,0 p. Equaion () an also wrin as; ( ln( p)) Q( p) (3) Susiu quaion (3) ino quaion (); Qp ( ) Q( p) ( p)( ln( p)) Equaion (3) an also simplifid as; Q( p) ln( p) Susiu quaion (5) ino quaion (4); ( Q( p) ) Q( p) ( p)( Q( p)) (4) (5) (6) ISBN: ISSN: (Prin); ISSN: (Onlin) WCECS 07

3 Prodings of h World Congrss on Enginring and Compur Sin 07 Vol II WCECS 07, Oor 5-7, 07, San Franiso, USA ( Q( p)) Q( p) ( p) (7) Q(0.) ( ln(0.9)) (8) Th diffrnial quaions an only oaind for pariular valus of, and. Whn, quaion (7) oms; Q a( p) ( p) (9) ( p) Q a( p) 0 (30) Whn, quaion (7) oms; Q ( p) ( p)( Q ( p)) ( p)( Q( p)) Q ( p) 0 (3) (3) Whn 3, quaion (7) oms; 3 Q ( p) 3( p)( Q ( p)) 3 (33) 3( p)( Q ( p)) Q ( p) 0 (34) IV. SURVIVAL FUNCTION Th survival funion of h 3- paramr Wiull disriuion is givn as; S ( ) (35) To oain h firs ordr ordinary diffrnial quaion for h survival funion of h 3-paramr Wiull disriuion, diffrnia quaion (35), o oain; S( ) (36) 0,,, 0. S( ) S( ) Th diffrnial quaions an only oaind for pariular valus of, and. Whn, quaion (37) oms; (37) S a( ) Sa( ) (38) S a( ) Sa( ) 0 (39) Whn, quaion (37) oms; S ( ) S( ) (40) S ( ) ( ) S( ) 0 (4), quaion (37) oms; Whn 3 3 S ( ) S( ) (4) 3 S( ) 3( ) S ( ) 0 (43) Equaion (37) is diffrniad in ordr o oain a simplifid ordinary diffrnial quaion; S( ) S( ) S( ) (44) ( ) S( ) S( ) S( ) (45),,, 0. Th following quaions oaind from (37) ar ndd o simplify quaion (45); S() S () ( ) ( ) S( ) S () ( ) ( ) S( ) ( ) S( ) Susiu quaions (46) and (48) ino quaion (45); S ( () ) ( ) S ( S ) S( ) ( ) S( ) (46) (47) (48) (49) Th sond ordr ordinary diffrnial quaion for h survival funion of h 3-paramr Wiull disriuion is givn y; ( ) S( ) S( ) ( ) S ( ) ( ) S( ) 0 (50) S(0) (5) S(0) (5) Alrnaivly, h ordinary diffrnial quaions an drivd using h rsuls oaind from h proailiy dnsiy funion. Equaion (3) an modifid using quaion (36) o oain; S( ) S( ) Whn, quaion (53) oms; (53) S d( ) S d ( ) (54) S ( ) S ( ) 0 (55) d d ISBN: ISSN: (Prin); ISSN: (Onlin) WCECS 07

4 Prodings of h World Congrss on Enginring and Compur Sin 07 Vol II WCECS 07, Oor 5-7, 07, San Franiso, USA Whn, quaion (53) oms; ( ) S ( ) S ( ) (56) ( ) S ( ) ( ( ) ) S ( ) 0 (57) Whn 3, quaion (53) oms; 3( ) S f( ) S ( ) 3 f (58) ( ) S ( ) ( 3( ) ) S ( ) 0 (59) f V. INVERSE SURVIVAL FUNCTION Th invrs survival funion of h 3- paramr Wiull disriuion is givn as; Qp ( ) ln (60) p To oain h firs ordr ordinary diffrnial quaion for h invrs survival funion of h 3-paramr Wiull disriuion, diffrnia quaion (60), o oain; Q( p) ln p p f (6) ln p Q( p) (6) pln p, 0,0 p. Equaion (60) an also wrin as; ln Qp ( ) (63) p ( Q( p) ) ln (64) p Susiu quaions (63) ino quaion (6); Qp ( ) Q( p) pln p Susiu quaion (64) ino quaion (65); ( Q( p) ) Q( p) p( Q( p) ) (65) (66) ( Q( p) ) Q( p) p (67) pq( p) ( Q( p) ) 0 (68) Q(0.) ln0 (69) Th diffrnial quaions an only oaind for pariular valus of, and. Som ass ar onsidrd and shown in Tal. Tal : Som lass of ODE for diffrn paramrs of h invrs survival funion of h 3-paramr Wiull disriuion Ordinary diffrnial quaion - pq( p) 0 - pq( p) pq( p) 3 0 p( Q( p) ) Q( p) 0 p( Q( p) ) Q( p) 0 p( Q( p) ) Q( p) 0 p( Q( p) ) Q( p) 0 VI. HAZARD FUNCTION Th hazard funion of h 3- paramr Wiull disriuion is givn as; h () (70) To oain h firs ordr ordinary diffrnial quaion for h hazard funion of h 3-paramr Wiull disriuion, diffrnia quaion (70), o oain; ( ) () h (7),,, 0. ( ) h( ) h( ) (7) Th firs ordr ordinary diffrnial quaion for h hazard funion of h 3-paramr Wiull disriuion is givn y; h( ) ( ) h( ) 0 (73) h(0) (74) To oain h sond ordr ordinary diffrnial quaion for h hazard funion of h 3-paramr Wiull disriuion, diffrnia quaion (7); 3 ( )( ) () 3 h (75),,, 0. Two ordinary diffrnial quaions an oaind from h simplifiaion of quaion (75); ODE ; Us quaion (70) in quaion (75); ( )( ) h( ) h( ) ( ) (76) ISBN: ISSN: (Prin); ISSN: (Onlin) WCECS 07

5 Prodings of h World Congrss on Enginring and Compur Sin 07 Vol II WCECS 07, Oor 5-7, 07, San Franiso, USA ( ) h( ) ( )( ) h( ) 0 (77) ODE ; Us quaion (7) in quaion (75) ( ) h ( ) h( ) ( ) (78) ( ) h ( ) ( ) h( ) 0 (79) ( ) h (0) (80) To oain h hird ordr ordinary diffrnial quaion for h hazard funion of h 3-paramr Wiull disriuion, diffrnia quaion (75); 4 ( )( )( 3) () 4 h (8),,, 0. Thr ordinary diffrnial quaions an oaind from h simplifiaion of quaion (8); ODE ; Us quaion (70) in quaion (8); ( )( )( 3) h( ) h( ) 3 ( ) (8) 3 ( ) h( ) ( )( )( 3) h( ) 0 (83) ODE ; Us quaion (7) in quaion (8); ( )( 3) h ( ) h( ) ( ) (84) ( ) h ( ) ( )( 3) h( ) 0 (85) ODE 3; Us quaion (75) in quaion (8); ( 3) h ( ) h ( ) ( ) (86) ( ) h ( ) ( 3) h ( ) 0 (87) ( )( ) h(0) 3 3 VII. REVERSED HAZARD FUNCTION (88) Th rvrsd hazard funion of h 3- paramr Wiull disriuion is givn as; j () (89) To oain h firs ordr ordinary diffrnial quaion for h rvrsd hazard funion of h 3-paramr Wiull disriuion, diffrnia quaion (89), o oain; j ( ) j( ) ( ) ( ) (90),,, 0. j( ) j( ) ( ) j( ) j( ) j( ) (9) (9) Th diffrnial quaions an only oaind for pariular valus of, and. Whn, quaion (9) oms; j a( ) ja( ) ja( ) (93) j a( ) ja( ) ja ( ) 0 (94), quaion (9) oms; Whn j ( ) j ( ) j( ) ( ) j( ) ( ( ) ) j ( ) ( ) j ( ) 0 Equaion (9) is diffrniad o oain; j( ) j( ) j( ) ( ) j ( ) j( ) ( ) (95) (96) (97),,, 0. Th following quaions oaind from (9) ar ndd o simplify quaion (97); ISBN: ISSN: (Prin); ISSN: (Onlin) WCECS 07

6 Prodings of h World Congrss on Enginring and Compur Sin 07 Vol II WCECS 07, Oor 5-7, 07, San Franiso, USA j( ) j () j() j( ) j () (98) (99) ( ) j( ) j () j() (00) ( ) j( ) j () j() (0) Susiu quaions (98) and (0) ino quaion (97); j () ( ) j( ) j( ) j () j( ) j( ) j( ) j() j ( ) ( ) j( ) j( ) j( ) j( ) j( ) ( ) ( ) j ( ) ( ) j ( ) (0) (03) Th ODEs an oaind for h pariular valus of h disriuion. Svral analyi, smi-analyi and numrial mhods an applid o oain h soluions of h rspiv diffrnial quaions [37-5]. Also omparison wih wo or mor soluion mhods is usful in undrsanding h link wn ODEs and h proailiy disriuions. VIII. CONCLUDING REMARKS In his work, diffrniaion was usd o oain som lasss of ordinary diffrnial quaions for h proailiy dnsiy funion (PDF), quanil funion (QF), survival funion (SF), invrs survival funion (ISF), hazard funion (HF) and rvrsd hazard funion (RHF) of h 3- paramr Wiull disriuion. In all, h paramrs ha dfin h disriuion drmin h naur of h rspiv ODEs and h rang drmins h xisn of h ODEs. ACKNOWLEDGMENT Th auhors ar unanimous in appriaion of finanial sponsorship from Covnan Univrsiy. Th onsruiv suggsions of h rviwrs ar graly appriad. REFERENCES [] A.M. Razali, A.A. Salih and A.A. Mahdi, Esimaion auray of Wiull disriuion paramrs, J. Appl. Si. Rs., vol. 5, no.7, pp , 009. [] M. Timouri and A.K. Gupa, On h Thr-paramr Wiull disriuion shap paramr simaion, J. Daa Si., vol., pp , 03. [3] G.W. Cran, Momn simaors for h 3-paramr Wiull disriuion, IEEE Trans. Rliailiy, vol. 37, pp , 988. [4] A. Adaia and L.K. Chan, Rous simaors of h 3-paramr Wiull disriuion, IEEE Trans. Rliailiy, vol. 34, no. 4, pp , 985. [5] D. Kundu and M.Z. Raqa, Esimaion of R = P(Y < X) for hrparamr Wiull disriuion, Sa. Pro. L., vol. 79, pp , 009. [6] G.H. Lmon, Maximum liklihood simaion for h hr paramr Wiull disriuion asd on nsord sampls, Thnomris, vol. 7, no., pp , 975. [7] M. Sirvani and G. 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