The Kumaraswamy Generalized Power Weibull Distribution

Transcription

1 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) The Kumrswmy Geerlized Power Weibull Distributio Mhmoud Ali Selim * d Abdullh Mhmoud Bdr Deprtmet of Sttistics, Fculty of Commerce, Al-Azher Uiversity, Egypt & Kig Khlid Uiversity, Sudi Arbi Abstrct A ew fmily of distributios clled Kumrswmy-geerlized power Weibull (Kgpw) distributio is proposed d studied. This fmily hs umber of well kow sub-models such s Weibull, expoetited Weibull, Kumrswmy Weibull, geerlized power Weibull d ew sub-models, mely, expoetited geerlized power Weibull, Kumrswmy geerlized power expoetil distributios. Some sttisticl properties of the ew distributio iclude its momets, momet geertig fuctio, qutile fuctio d hzrd fuctio re derived. I dditio, mximum likelihood estimtes of the model prmeters re obtied. A pplictio s well s comprisos of the Kgpw d its sub-distributios is give. Keywords: Geerlized power Weibull distributio, Kumrswmy distributio, Mximum likelihood estimtio, Momet geertig fuctio, Hzrd rte fuctio.. Itroductio I relibility models, the probbility distributios re most ofte used s time to filure distributios. I sme cotext, the relibility model qulity sigifictly depeds o the success i selectig pproprite probbility distributio of the pheomeo uder discussio. Durig the pst decdes, specific group of the clssicl distributios such s, expoetil, Weibull d Ryleigh distributios were used for modelig lifetime dt. However, i prctice, we fid tht most of these distributios re ot flexible eough to ccommodte differet pheome. For this reso, the sttisticis hve worked o developmet d exted of these distributios to become more flexible d more suited for modelig dt i prctice. The trditiol Weibull distributio by Wloddi Weibull (95) is oe of the most used lifetime distributios for modelig lifetime dt. However, the Weibull distributio does ot provide o-mootoe filure rtes tht re commo i relibility d survivl lysis. My versios of geerlized Weibull distributio hve rise out of the eed to improve its properties. The first geerliztio of Weibull distributio provides bthtub shped hzrd rte is the expoetited Weibull distributio due to Mudholkr et l. (995). The expoetited Weibull distributio c be used quite effectively to lyze the lifetime dt i plce of Weibull distributio. Also, Nikuli d Hghighi (26) proposed ew geerliztio of the Weibull distributio by itroducig dditiol shpe prmeter, which they clled the geerlized power Weibull distributio. The rdom vrible X hs the geerlized power Weibull (gpw) distributio if its cdf d pdf re G gpw (x) = exp { ( + ( x λ ) α) }, α, λ, >, x > () d g gpw (x) = α λ α xα ( + ( x α λ ) ) exp { ( + ( x α λ ) ) }, α, λ, >, x > (2) where λ is scle prmeter d α d re two shpe prmeters. It is reduced to the stdrd Weibull distributio whe, =. Nikuli d Hghighi (26) showed tht the hzrd rte fuctio of the geerlized power Weibull distributio hs ice d flexible properties d c be costt, mootoe d o-mootoe shped. This distributio is ofte used for costructig ccelerted filures times models tht describe depedece of the lifetime distributio o expltory vribles. They lso illustrted tht the gpw provides good fit to the well-kow rdomly cesored survivl times dt for ptiets t rm A of the hed-d-eck ccer cliicl tril by usig chi-squred goodess-of-fit test. Eugee et l. (22) itroduced the bet-geerted fmily to geerlize the cotiuous probbility distributio s follows

2 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) F B (x) = B(, b) u ( u) b du (3) the correspodig pdf of the bet geerted distributio is f(x) = B(, b) g(x)g(x) [ G(x) b (4) where B(α, β) is the bet fuctio, g(x) d G(x) re the pdf d cdf for pret probbility distributio. The formul give i (4) hs bee used first by Eugee et l. (22) to geerte the bet orml distributio. After tht, umber of uthors proposed ew bet-geerted fmily of distributios iclude Fmoye et l. (25) itroduced the bet-weibull distributio, Ndrjh d Kotz (24) itroduced the bet Gumbel distributio Ndrjh d Kotz (26) itroduced the bet expoetil distributio, Kog et l. (27) itroduced the bet-gmm distributio, Cordeiro d Lemote (2) itroduced bet Lplce distributio. Mmeli d Musio (23) itroduced the bet skew-orml distributio. Recetly, Merovci d Shrm (24) itroduced bet-lidley distributio, Jfri et l. (24) itroduced bet-gompertz distributio, Chukwu d Ogude (25) itroduced Bet Mekhm distributio d MirMostfee et l. (25) itroduced bet Lidley distributio. For good review of bet- geerted distributios, oe my refer to Lee et l. (23). Kumrswmy (98) proposed two-prmeter distributio o (, ), so-clled Kumrswmy distributio, d deoted by Kum(, b). Its cumultive distributio fuctio (cdf) is G(x) F kum (x) = ( x ) b, < x < (5) d its desity fuctio is f kum (x) = bx ( x ) b, < x < (6) Kum(, b) distributio, ccordig to Joes (29) like the bet distributio, c be uimodl, uitimodl, icresig, decresig or costt d hs dvtge over the bet distributio tht, Kum(, b) distributio does ot ivolve y specil fuctio such s the bet fuctio d its cumultive distributio fuctio hs simple closed form. For this resos, Cordeiro d de Cstro (2) developed the bet-geerted fmily by employig the Kumrswmy distributio isted of bet distributio. For rbitrry bselie cdf G(x), Cordeiro d de Cstro (2) defied the cdf d pdf of Kumrswmy geerlized distributios (Kum-G), respectively, s follow d F(x) = { G(x) } b (7) f(x) = bg(x)g(x) b { G(x) } b (8) where g(x) = dg(x)/dx d > d b > re two dditiol shpe prmeters of the G(x) distributio which role re to gover skewess d til weights of the geerted distributio. This type of geerliztios cotis distributios with uimodl d bthtub shped hzrd rte fuctios d hve some desirble structurl properties compred with bet-geerted fmily of distributios, for detil see Cordeiro d de Cstro (2) d Joes (29). Severl geerlized distributios from (5) hve bee studied i the literture icludig, the Kw-Weibull distributio by Cordeiro et l. (2), Kw-Gumbel distributio by Cordeiro et l. (2), Kw-geerlized gmm distributio by Psco et l. (2), Kw-log-logistic distributio by Tigo et. l. (22), Kw-modified Weibull distributio by Cordeiro et. l. (22), Recetly, Gosh (24) itroduced Kw-hlf-Cuchy distributio, Atoio t. el. (24) itroduced Kw-geerlized Ryleigh distributio d (Roch et l. 25) itroduced Kw- Gompertz distributio. I this pper, we pply the works of Kumrswmy (98), Cordeiro d de Cstro (2), Nikuli d Hghighi (26) i order to study the mthemticl properties of ew distributio referred to s the Kumrswmy geerlized power Weibull (Kgpw) distributio. The rest of the rticle is orgized s follows. Sectio 2 itroduces the Kumrswmy geerlized power Weibull distributio. Some sttisticl properties of

3 f(x) Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) the Kumrswmy geerlized power Weibull distributio re discussed i Sectio 3. The pdf of order sttistics of the Kgpw model is itroduced i Sectio 4. Mximum likelihood estimtio is ivestigted i Sectio 5. I Sectio 6, rel dt set re used to illustrte the usefuless of the Kgpw model. Cocludig commets re give i Sectio The Kumrswmy Geerlized Power Weibull Distributio I this sectio, we itroduce the pdf d the cdf of Kgpw distributio by settig the gpw bselie fuctios () d (2) i Equtios (5) d (6), the the cdf d pdf of the Kgpw distributio re obtied s follow b F kgpw (x) = [ ( e (+(x λ )α ) ),, b, α, λ, >, x > (9) d f kgpw (x) = bα λ α xα ( + ( x α λ ) ) e (+(x λ )α ) [ e (+(x λ )α ) b { [ e (+(x λ )α ) } () where λ is scle prmeter d the others prmeters, b, α d re shpe prmeters. The possible shpes of the pdf d cdf of Kgpw distributio re provided for five combitios of the prmeters i Figure d Figure 2, respectively. The shpes i Figure, show tht the pdfs of Kgpw distributio c be mootoiclly decresig or positively skewed =.5, b=5, =.5, =.4, =2 =.5, b=.5, =.6, =7, =2 =.2, b=, =.6, =.2, = =.9, b=., =3, =.2, =2 =5, b=.5, =3, =.5, = Figure (): Some Possible Shpes of the Kgpw Desity Fuctio x 2

4 F(x) Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) =.5, b=5, =.5, =.4, =2.2 =.5, b=.5, =.6, =7, =2 =.2, b=, =.6, =.2, =. =.9, b=., =3, =.2, =2 =5, b=.5, =3, =.5, = Figure (2): Some Possible Shpes of the Kgpw Cumultive Desity Fuctio 2. Some sub-models of the Kgpw The Kgpw distributio is very flexible seeig s this distributio icludes severl well-kow distributios s sub-models bsed o specil vlues of the prmeters (, b, d α). These sub-models re ) Settig = b =, we obti geerlized power Weibull distributio (Nikuli d Hghighi 26) with cdf: F(x) = e (+(x λ )α ), 2) Settig b =, we obti expoetited geerlized power Weibull distributio (ew) with cdf: F(x) = ( e (+(x λ )α ) ), 3) Settig =, we obti Kumrswmy Weibull distributio distributio (cordiro et. l. 2) with cdf: F(x) = ( ( e (x λ )α ) ), 4) Settig α = we obti Kumrswmy geerlized power expoetil distributio (ew) with cdf: F(x) = ( ( e (+x λ ) ) ), 5) Settig b = =, we obti expoetited Weibull distributio (Mudholkr et l.(995)) with cdf: F(x) = ( e (x λ )α ), 6) Settig = = b = we obti Weibull distributio with cdf: F(x) = e (x λ )α, 7) Settig α =, b = we obti expoetited geerlized power expoetil distributio (ew) with cdf: F(x) = ( e (+x λ ) ), 8) Settig α = = we obti Kumrswmy expoetil distributio with cdf: F(x) = ( ( e x λ) ), 9) Settig b = = α =, we obti expoetited expoetil distributio (Gupt d Kudu (2)) with cdf: F(x) = ( e x λ), ) Settig α = = b =, we obti the expoetil extesio distributio (Ndrjh d Hghighi (2)) with cdf: F(x) = e (+x λ ), ) Settig α = = = b =, we obti expoetil distributio with cdf: x F(x) = e x λ, b b b 3

5 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) 2) Settig =, α = 2, we obti Kumrswmy Burr type X distributio (NEW) with cdf: F(x) = ( ( e (x λ )2 ) ), 3) Settig b = =, α = 2 we obti Burr type X distributio with cdf: F(x) = ( e (x λ )2 ). 2.2 Expsios for the cumultive d desity fuctios The expsio for the cumultive distributio fuctio of Kgpw c be derived by usig the geerlized biomil theorem. For y rel umber r > d z < the biomil expsio is ( z) r = ( ) i ( r i ) zi i= where ( r i ) = r(r ) (r i+). i! Usig the biomil expsio () i equtio (9), we get the cdf s power series expsio s follows F(x) = p i G gpw (x) i (2) i= where p i = ( ) i ( b i ) d G gpw(x) deotes the gpw cumultive distributio with prmeters α, λ d. Which mes tht, G gpw (x) i deotes the cdf of expoetited geerlized power Weibull (egpw) with prmeters α, λ, d i. Usig the biomil expsio (), gi i the lst term of (2), we get F(x) = ( ) i+j ( b i ) (i j ) i= j= e j( (+(x λ ) α ) ) Differetitig (3) with respect to x gives the expsio of pdf s follow f kgpw (x) = α λ α ( )i+j ( b i ) (i j ) j i= j= x α ( + ( x α) λ ) e j( (+(x λ ) α ) ) 2.3 The hzrd d survivl fuctios Filure rtes or hzrd rtes re importt subject i the idustry, egieered system, fice d fudmetl to the pl of socil security, medicl isurce d sfe systems i wide vriety of pplictios. The hzrd rte fuctio (hrf) of the rdom vrible T tht hs the Kgpw is give by h(t) = bαt α ( + ( t λ )α ) e (+(t λ )α ) [ e (+(t λ )α ) λ α ( ( e (+(t λ )α ) ) ) b () (3) (4), t > (5) Usig the expsios i (3) d (4), the hrf of the Kgpw distributio i (5) c be expressed i the mixture form s follows α ( ) i+j ( b j= i ) (i j ) j t α ( + ( t i= λ )α ) h(t) = Note tht for ll b,, λ we hve λ α ( ) i+j ( b i= j= i ) (i j ) e j( (+(t λ ) α ) ) e j( (+( t λ )α ) ), t > 4

6 h(x) Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) for, α > b h() = { for = α = λ for, α < The hzrd rte fuctio of Kgpw distributio c be hve vrious shpes, icludig costt, mootoiclly decresig or icresig, bthtub d upside dow bthtub. More specificlly, the hzrd rte curve is () mootoe icresig if either α > d α > or α = d >, (b) mootoe decresig if either < α < d α < or < α < d α =, (c) uimodl (iverted bthtub shped) if α > d < α <, (d) bthtub shped if < α < d α >, (e) costt, h(t) = b if = α = =. λ Figure 3, provides plots of the hzrd fuctio of Kgpw distributio for some selected prmeters vlues. These plots show flexibility of hzrd rte fuctio tht mkes the Kgpw hzrd rte fuctio useful d suitble for o-mootoe hzrd behviors tht re more likely to be observed i rel life situtios. I relibility theory there re severl importt fuctios such s the survivl fuctio s(t), reverse hzrd fuctio r(t) d the cumultive hzrd rte fuctio H(t). These fuctios correspodig of the Kgpw distributio, tke the followig forms: s(t) = F(x) = ( ( e (+(t λ )α ) ) ) b, t >, (6) bαt α ( + ( t b λ )α ) e (+(t λ )α ) [ e (+(t λ )α ) { [ e (+(t λ )α ) } r(t) = b λ α [ ( ( e (+(t λ )α ) ) ) d t > (7) H(t) = b l ( ( e (+(t λ )α ) ) ), t > (8) respectively =.5, b=5, =.5, =.4, =2 =.5, b=.5, =.6, =7, =2 =.2, b=, =.6, =.2, = =.9, b=., =3, =.2, =2 =5, b=.5, =3, =.5, = Figure (3): Some Possible Shpes of the Kgpw Hzrd Rte Fuctio x 5

7 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) 3. The Sttisticl Properties of Kgpw Distributio I this sectio, we preset the qutile fuctio, the momets d the momet geertig fuctio, skewess, kurtosis d rdom vribles geertio for the Kgpw distributio. 3. Qutile fuctio There re severl mesures for loctio d dispersio such s medi, the iterqurtile rge, the qurtiles, the skewess d the kurtosis c be obtied by usig the qutile fuctio. The defiitio of the q-th qutile is the rel solutio of the followig equtio F(x q ) = q, where q Thus, the qutile fuctio Q(q) correspodig of the Kgpw distributio is Q(q) = λ {( l [ ( ( q) b) ) } The medi M(x) of Kgpw distributio c be obtied from previous fuctio, by settig q =.5, s follows M(X) = λ {( l [ ( (.5) b) ) } Also, the qurtiles of the Kgpw distributio c be obtied by puttig q =.25 d q =.75 i (9). 3.2 Skewess d kurtosis The sttisticl mesures of skewess d kurtosis ply importt role i describig shpe chrcteristics of the probbility distributios. The Bowley s skewess mesure bsed o qurtiles (Keey d Keepig, (962)) is give by α α (9) (2) Sk = Q(3 4 ) 2Q( 2 ) + Q( 4 ) Q( 3 4 ) Q( 4 ) (2) d the Moors kurtosis mesure bsed o octiles (Moors (988)) is give by Ku = Q(7 8 ) Q(5 8 ) + Q(3 8 ) Q( 8 ) Q( 6 8 ) Q(2 8 ) (22) The previous mesures Sk d Ku hve umber of dvtges compred to the clssicl mesures of skewess d kurtosis, e.g. they re less sesitive to outliers d they exist for the distributios eve without defied the momets. 3.3 Rdom vribles geertio The qutile fuctio of the Kgpw hs closed form, which mkes the simultio from this distributio esier. Whe the prmeters, b, α, λ d re kow, we c geerte Kgpw rdom vribles from the qutile fuctio (9) s follows X = λ {( l [ ( ( u) b) ) } where, u is geerted umber from the Uiform distributio (, ). α (23) 6

8 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) 3.4 Momets d momet geertig fuctio If X hs the Kgpw distributio, the momets d momet geertig fuctio re give by the followig theorems Theorem. If X is rdom vrible hvig the pdf (2), For iteger vlue of r α, the r-th momet bout zero c be determied s E(X r ) = λ r φ ijk e j j k Γ ( k +, j) (24) r i where, φ ijk = α ( ) i+j+r α k ( b k= i ) ( i j= j ) ( r b α k ). Proof. From defiitio, the r-th momet of Kgpw distributio is E(X r ) = x r f kgpw (x)dx (25) By settig f Kgpw (x) from (4) i previous equtio yields b (i+) E(X r ) = α λ α ( )i+j ( b i ) ( i j ) j x r+α ( + ( x α) λ ) i= j= e j( (+(x λ ) α ) Substitutio v = + ( x λ )α i the previous equtio d fter tht usig the biomil expsio, we get b i r α E(X r ) = λ r ( ) i+j+r α k ( b i ) ( i j ) ( r α j= k= k ) jej v k e jv The lst term is Γ ( k +, j) jk +. Therefore, (27) c be reduced to (24). Theorem 2. If X is rdom vrible hvig the pdf (2), for iteger vlue of r α the momet geertig fuctio is M x (t) = λ r tr e j Γ ( k +, j) φ r! ijk r= Proof. The momet geertig fuctio of Kgpw distributio is give by M x (t) = e tx j k f kgpw (x)dx Usig the fct tht e tx = r=, we get (tx) r r! dv ) dx (26) (27) M x (t) = tr E(X r ) (29) r! r= Isertig (24) i equtio (29) yields the mgf of Kgpw i (28). The umericl vlues of me d vrice for vrious choices of prmeters re give i Tble d Tble 2, respectively. (28) 7

9 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) Tble : Me of the Kgpw Distributio for Some Vlues of, b,, α d λ = 3 =.5 =2.5 =3.5 b α=.5 α=2.5 α=3.5 α=.5 α=2.5 α=3.5 α=.5 α=2.5 α= Tble idictes tht, the me of Kgpw distributio is decresig whe b icresig with fixed the others prmeters. While, the me of Kgpw distributio is icresig whe icresig or α with fixed the others prmeters. 8

10 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) Tble 2: Vrice of the Kgpw Distributio for Some Vlues of, b,, α d λ= 3 =.5 =2.5 =3.5 b α=.5 α=2.5 α=3.5 α=.5 α=2.5 α=3.5 α=.5 α=2.5 α= Tble 2 idictes tht, the vrice of Kgpw distributio will decrese with icrese the prmeters vlues. 4. Order Sttistics Suppose X (), X (2),, X () deote the order sttistics of rdom smple of size drw from cotiuous distributio with cdf F(x) d pdf f(x), the the pdf of X (l) is give by 9

11 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) f l: (x) =! ( l)! (r l)! f(x)[f(x)l [ F(x) l (3) New, If X is rdom vrible followig Kgpw distributio the, by substitutig F(x) d f(x) i eqs (9) d () i to eq (3), we get the Kgpw desity of the l-th order sttistics s follows f l: (x) =! bα ( l)! (l )! λ α xα ( + ( x α λ ) ) e (+(x λ )α ) [ e (+(x λ )α ) b l b( l+) { [ [ e (+(x λ )α ) } { [ e (+(x λ )α ) } (3) whe l = d whe l =, the pdf of order sttistics become d f : (x) = bα λ α xα ( + ( x α λ ) ) e (+(x λ )α ) [ e (+(x λ )α ) b { [ e (+(x λ )α ) } f : (x) = bα λ α xα ( + ( x α λ ) ) e (+(x λ )α ) [ e (+(x λ )α ) b b { [ [ e (+(x λ )α ) } { [ e (+(x λ )α ) } (32) (33) respectively. 5. Mximum Likelihood Estimtio I this sectio, we determie the mximum likelihood estimtes (MLEs) for the prmeters σ = (, b, α, λ, ) of the Kgpw distributio. Let x, x,, x be complete rdom smple of size from the Kgpw distributio. The likelihood fuctio (LF) is give by L(σ x) = ( bα λ α ) x α i ( + ( x α i λ ) { [ e (+(x i λ )α ) } ) e (+(x i λ )α ) [ e (+(x i λ )α ) b (34) d the log-likelihood fuctio (logl) is give by logl = l ( bα λ α ) + + (α ) l x i +( ) l ( e (+(x i λ )α ) ) + ( ) l ( + ( x α i λ ) ) ( + ( x α i λ ) + (b ) l [ ( e (+(x i λ )α ) The derivtives of the logl with respect to the ukow prmeters, b, α, λ d re ) ) (35) l L = + l( ω,α,λ) (b ) l( ω,α,λ) ( ω,α,λ ) ( ω,α,λ ) (36) 2

12 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) l L b = b + l( ( ω,α,λ) ) l L α = α l λ + l x i + ( ) l ( λ ) (x i l L λ x i φ α,λ λ )α l ( x α i λ ) (x i λ ) ω,α,λ φ α,λ (37) { ( )ω,α,λ + (b )( ω,α,λ) ω,α,λ ( ω,α,λ ) } (38) = α λ + α λ α+ [x i α φ α,λ α( ) λ α+ α [ x i + α φ α,λ λ α+ x i α φ α,λ ( ) ω,α,λ { ω,α,λ + (b )( ω,α,λ) ( ω,α,λ ) } (39) l L = + l(φ α,λ) l (φ α,λ ) φ α,λ + ( ) l(φ α,λ) ω,α,λ φ α,λ ω,α,λ (b ) l(φ α,λ) ω,α,λ φ α,λ ( ω,α,λ ) where, ω,α,λ = e (+(x i λ )α ) l L ( ω,α,λ ), φ α,λ = ( + ( x i λ )α ). The mximum likelihood estimtes of, b, α, λ d re the simulteous solutios of the equtios l L =, b =, l L α =, l L λ (4) =, l L =. These equtios cot be solved lyticlly d sttisticl softwre c be used to solve them umericlly by usig itertive techiques like the Newto-Rphso lgorithm. 6. Applictio I this sectio, we hve give pplictio of Kgpw distributio usig rel dt set to illustrte tht Kgpw distributio provides sigifict improvemets over its sub-models Weibull (W) d geerlized power Weibull (gpw). The rel dt set is tke from Bdr d Priest (982). The dt represet the stregth dt mesured i GPA, for sigle crbo fibers were tested uder tesio t guge legths of,, 2 d 5 mm. For illustrtive purpose, we cosider oly the dt set cosistig the sigle fibers of 2 mm, with smple of size 63. The dt re:.9, 2.32, 2.23, 2.228, 2.257, 2.35, 2.36, 2.396, 2.397, 2.445, 2.454, 2.474, 2.58, 2.522, 2.525, 2.532, 2.575, 2.64, 2.66, 2.68, 2.624, 2.659, 2.675, 2.738, 2.74, 2.856, 2.97, 2.928, 2.937, 2.937, 2.977, 2.996, 3.3, 3.25, 3.39, 3.45, 3.22, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.48, 3.435, 3.493, 3.5, 3.537, 3.554, 3.562, 3.628, 3.852, 3.87, 3.886, 3.97, 4.24, 4.27, 4.225, 4.395, 5.2. I Tble 3. the mximum likelihood estimtes of the ukow prmeters of the Kgpw, gpw d Weibull distributios re give, log with the criteri likelihood ( l (k)), Akike iformtio criterio (AIC), corrected Akike iformtio criterio (CAIC) d H-Qui criterio (HQC) (see H d Qui (978)). These criteri tke the followig forms, AIC = 2ll (k) + 2k, AICC = 2 AIC + 2k(k+) d k HQC = 2ll (k) + 2kl(l()) where, l (k) be the mximum likelihood of model with umber of prmeters k bsed o smple of size. Also, the plots of the empiricl d estimted cdf s of these distributios re give i Figure (4) s grphicl illustrtio of the goodess of fit for these dt. Tble 3: The estimted prmeters d sttistics l (k), AIC, AICC d HQC for fitted models Estimtes Sttistics Model b α λ l (k) AIC AICC HQC Weibull gpw Kgpw

13 F(x) f(x) Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) It is observed from Tble 3 tht l (k), AIC, AICc d HQC re lowest i cse of Kgpw distributio. Therefore, we c coclude tht Kgpw distributio performs better th Wibull d gpw distributios. The figures (4) d (4b) lso cofirm good fit of the Kgpw model for the dt set..3.2 () Kgpw gpw W x (b) Empiricl Kgpw gpw W x Figure 4: () Estimted pdfs of the Kgpw Distributio d its Sub-Models for the Stregth Dt from Bdr d Priest (982). (b) Empiricl d Estimted cdfs of the Kgpw Distributio d its Sub-Models for the Stregth Dt from Bdr d Priest (982). 7. Cocludig Remrks I this rticle, we defie ew model, which is clled the Kumrswmy geerlized power Weibull distributio. The ew distributio geerlizes the geerlized power Weibull distributio defied by Nikuli d Hghighi (26). Some mthemticl properties re derived d plots of the pdf, cdf d hzrd fuctio re preseted to show the flexibility of the ew distributio. The mximum likelihood estimtio for the model prmeters is discussed. Filly, pplictio of the proposed model to rel dt set is give to illustrte tht Kgpw distributio c be used quite effectively to provide better fits th other vilble models. 22

14 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) Refereces. Atoio E. Gomes, Cibele Q. d-silv, Guss M. Cordeiro d Edwi M.M. Orteg (24). A ew lifetime model: the Kumrswmy geerlized Ryleigh distributio, Jourl of Sttisticl Computtio d Simultio, 84(2), Bder, M. G., d Priest, A. M. (982). Sttisticl spects of fibre d budle stregth i hybrid composites, Progress i sciece d egieerig of composites, Chukwu, A., d A. Ogude. (25). O the bet Mekhm distributio d Its pplictios. Americ Jourl of Mthemtics d Sttistics 5 (3): Cordeiro, G.M. d Lemote, A.J. (2). The bet Lplce distributio, Sttistics d Probbility Letters, 8, Cordeiro, G.M., de Cstro, M. (2). A ew fmily of geerlized distributios, Jourl of Sttisticl Computtio d Simultio, 8, Cordeiro, G.M., Orteg, E.M.M. d Ndrjh, S. (2). The Kumrswmy Weibull distributio with pplictio to filure dt, Jourl of the Frkli Istitute, 347, Cordeiro, G.M., Orteg, E.M.M. d Silv, G.O. (22). The Kumrswmy modified Weibull distributio: theory d pplictios, Jourl of Sttisticl Computtio d Simultio, DOI:.8/ , (prit). 8. Cordeiro, GM, Ndrjh, S d Orteg, E.M.M. (2). The Kumrswmy Gumbel distributio, Sttisticl Methods d Applictios, 2(2), Eugee, N., Lee, C., Fmoye, F. (22). Bet-orml distributio d its pplictios, Commuictios i Sttistics - Theory d methods, 3(4), Fmoye, F., Lee, C. d Olumolde, O. (25). The bet-weibull distributio, Jourl of Sttisticl Theory d Applictios, 4(2), Ghosh, I. (24). The Kumrswmy-hlf-Cuchy distributio: properties d pplictios, Jourl of Sttisticl Theory d Applictios, 3(2), Gupt, R. D. d Kudu, D. (2). Expoetited expoetil fmily; ltertive to gmm d Weibull, Biometricl Jourl, 33(), H, E. J., d B. G. Qui (978),The determitio of the order of utoregressio, Jourl of the Royl Sttisticl Society, B, 4, Jfri, A. A., S. Thmsebi, d M. Alizdeh. (24). The bet-gompertz distributio. Revist Colombi de Estdístic 37 (): Joes, M. C. (29). Kumrswmy s distributio: bet-type distributio with some trctbility dvtges, Sttisticl Methodology, 6(), Keey, J. F. d Keepig, E. S. (962). Mthemtics of Sttistics, Prt Oe. Third Editio, Priceto, New Jersey: V Nostrd. 7. Kog, L., Lee, C. d Sepski, J.H. (27). O the properties of bet-gmm distributio, Jourl of Moder Applied Sttisticl Methods, 6, Kumrswmy, P. (98). Geerlized probbility desity fuctio for double-bouded rdom processes, Jourl of Hydrology, 462, Lee, C., Fmoye, F., Alztreh. (23). A Methods for geertig fmilies of uivrite cotiuous distributios i the recet decdes, WIREs Comput. Stt., 5, Mmeli, V., Musio, M. (23). A geerliztio of the skew-orml distributio: the bet skew-orml, Commuictios i Sttistics-Theory d Methods, 42(2), Merovci, F., d V. K. Shrm. (24). The bet-lidley distributio: Properties d Applictios. Jourl of Applied Mthemtics, 24, MirMostfee, S., M. Mhdizdeh, d S. Ndrjh. (25). The bet Lidley distributio. Jourl of Dt Sciece 3 (3). 23. Mudholkr, G. S., Srivstv, D. K. d Freimer, M. (995). The expoetited Weibull fmily: A relysis of the bus-motor-filure dt, Techometrics, 37,

15 Mthemticl Theory d Modelig ISSN (Pper) ISSN (Olie) 24. Ndrjh S., Kotz, S. (24). The bet Gumbel distributio, Mthemticl Problems i Egieerig, 4, Ndrjh, S. d Kotz, S. (26). The bet expoetil distributio, Relibility Egieerig d System Sfety, 9, Nikuli, M., Hghighi, F. (26). A chi-squred test for the geerlized power Weibull fmily for the hed-d-eck ccer cesored dt, Jourl of Mthemticl Scieces, 33, Psco, A.R.M., Orteg, E.M.M. d Cordeiro, G.M. (2). The Kumrswmy geerlized gmm distributio with pplictio i survivl lysis, Sttisticl Methodology, 8, Roch, R., S. Ndrjh, V. Tomzell, F. Louzd, d A. Eudes. (25). New defective models bsed o the Kumrswmy fmily of distributios with pplictio to ccer dt sets. Sttisticl methods i medicl reserch: Tigo Vi Flor de St, Edwi M.M. Orteg, Guss M. Cordeiro d Giov O. Silv, (22). The Kumrswmy-log-logistic distributio, J. Sttist. Theor. d Applic., (3), Weibull, W. (95). A sttisticl distributio fuctio of wide pplicbility, J. Appl. Mech., 8, Mhmoud Ali Selim is ssistt Professor of Sttistics i the Deprtmet of Sttistics t Al-Azhr Uiversity. He received Ph.D. (2) i Sttistics from Al-Azhr Uiversity, Ciro, Egypt. He is ow o leve t Kig Khlid Uiversity, Sudi Arbi. His reserch iterests iclude probbility distributio, relibility, sttisticl ifereces ivolvig order sttistics d record vlues. 24