The Log-Kumaraswamy Generalized Gamma Regression Model with Application to Chemical Dependency Data

Transcription

1 Journal of Daa Scence ), The Log-Kumaraswamy Generalzed Gamma Regresson Model wh Applcaon o Chemcal Dependency Daa Marcelno A. R. Pascoa 1, Clauda M. M. de Pava 2, Gauss M. Cordero 3 and Edwn M. M. Orega 4 1 Unversdade Federal de Mao Grosso, 2 Unversdade Federal de São João del Re, 3 Unversdade Federal de Pernambuco and 4 Unversdade de São Paulo Absrac: The fve parameer Kumaraswamy generalzed gamma model Pascoa e al., 2011) ncludes some mporan dsrbuons as specal cases and s very useful for modelng lfeme daa. We propose an exended verson of hs dsrbuon by assumng ha a shape parameer can ake negave values. The new dsrbuon can accommodae ncreasng, decreasng, bahub and unmodal shaped hazard funcons. A second advanage s ha also ncludes as specal models recprocal dsrbuons such as he recprocal gamma and recprocal Webull dsrbuons. A hrd advanage s ha can represen he error dsrbuon for he log-kumaraswamy generalzed gamma regresson model. We provde a mahemacal reamen of he new dsrbuon ncludng explc expressons for momens, generang funcon, mean devaons and order sascs. We oban he momens of he log-ransformed dsrbuon. The new regresson model can be used more effecvely n he analyss of survval daa snce ncludes as submodels several wdely-known regresson models. The mehod of maxmum lkelhood and a Bayesan procedure are used for esmang he model parameers for censored daa. Overall, he new regresson model s very useful o he analyss of real daa. Key words: Censored daa, generang funcon, Kumaraswamy generalzed gamma dsrbuon, log-gamma generalzed regresson, momen, survval funcon. 1. Inroducon Sandard lfeme dsrbuons usually presen very srong resrcons o produce bahub curves, and hus appear o be unapproprae for daa wh hs characersc. The gamma dsrbuon s he mos popular model for analyzng skewed daa. The generalzed gamma GG) Sacy, 1962) dsrbuon ncludes Correspondng auhor.

2 782 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. as specal models he exponenal, Webull, gamma and Raylegh dsrbuons, among ohers. I s suable for modelng daa wh hazard rae funcon of dfferen forms ncreasng, decreasng, bahub and unmodal) and hen s useful for esmang ndvdual hazard funcons and boh relave hazards and relave mes Cox, 2008). The GG dsrbuon has been used n several research areas such as engneerng, hydrology and survval analyss. In fac, Orega e al. 2003) dscussed nfluence dagnoscs n GG regresson models, Nadarajah and Gupa 2007) appled hs dsrbuon o drough daa and Cox e al. 2007) presened a paramerc survval analyss and axonomy of GG hazard funcons. For X 1 and X 2 ndependen GG random varables, Al e al. 2008) derved he exac dsrbuons of he produc X 1 X 2 and quoen X 1 /X 2 and provded applcaons of her resuls o drough daa from Nebraska. Furher, Gomes e al. 2008) focused on parameer esmaon, Orega e al. 2008) compared hree ypes of devance componen resduals n GG regresson models under censored observaons, Cox 2008) dscussed and compared he F-generalzed famly wh he GG model, Almpands and Koropoulos 2008) presened a ex-ndependen auomac phone segmenaon algorhm based on hs dsrbuon and Nadarajah 2008a) analyzed some ncorrec references wh respec o s use n elecrcal and elecronc engneerng. More recenly, Barkauskas e al. 2009) suded he nose par of a specrum as an auoregressve movng average ARMA) model wh he nnovaons havng he GG dsrbuon and Malhora e al. 2009) provded a unfed analyss for wreless sysem over generalzed fadng channels ha s modeled by a wo parameer GG model. Also, Orega e al. 2009) proposed a modfed GG regresson model o allow he possbly ha long-erm survvors may be presened n he daa and Cordero e al. 2010) defned he exponenaed generalzed gamma EGG) dsrbuon. Ths dsrbuon due o s flexbly n accommodang many forms of he rsk funcon seems o be an mporan model ha can be used n a varey of problems n survval analyss. In he las years, new dsrbuons for modelng survval daa based on exensons of he Webull dsrbuon were developed. Mudholkar e al. 1995), Xe and La 1995), La e al. 2003) and Carrasco e al. 2008) nroduced he exponenaed Webull EW), addve Webull, modfed Webull MW) and generalzed modfed Webull GMW) dsrbuons, respecvely. Furher, he man movaon for he exponenaed generalzed gamma EGG) dsrbuon Cordero e al., 2010) s ha conans as specal cases he GG, EW, exponenaed exponenal EE) Gupa and Kundu, 1999) and generalzed Raylegh GR) Kundu and Raqab, 2005) dsrbuons. The Kumaraswamy generalzed gamma KGG) dsrbuon Pascoa e al., 2011) can model four ypes of he falure rae funcon.e. ncreasng, decreasng, unmodal and bahub) dependng on he values of s parameers. I s also

3 The Log-Kumaraswamy Generalzed Gamma Regresson Model 783 suable for esng goodness-of-f of some sub-models, such as he EGG, GG, EW, Webull and GR dsrbuons. In hs paper, we defne an exended form of he KGG dsrbuon o cope wh several recprocal ype dsrbuons and sudy some of s srucural properes. We exend he model by Pascoa e al. 2011), snce a shape parameer can ake negave values. The mehod of maxmum lkelhood s used for esmang he model parameers. Unless oherwse saed, all of he resuls presened n he paper are new and orgnal. I s expeced ha hey could encourage furher research on he new dsrbuon. Dfferen forms of regresson models have been proposed n survval analyss. Among hem, he locaon-scale regresson model Lawless, 2003) s dsngushed snce s frequenly used n clncal rals. Here, we propose a locaon-scale regresson model based on he new dsrbuon called he log-kumaraswamy generalzed gamma LKGG) regresson model, whch s a feasble alernave for modelng he four ypes of falure rae funcons. The res of he paper s organzed as follows. In Secon 2, we defne an exended verson of he KGG dsrbuon. In Secon 3, we derve s ordnary momens, momen generang funcon mgf), order sascs and her momens. In Secon 4, we defne he LKGG dsrbuon and derve an explc expresson for s momens. In Secon 5, we propose he LKGG regresson model for analyss of censored daa. We esmae he model parameers by maxmum lkelhood, derve he observed nformaon marx and dscuss a Bayesan mehodology o esmae he model parameers. In Secon 6, we llusrae he poenaly of he new regresson model by means of a real daa se n chemcal dependency. Secon 7 ends wh some concludng remarks. 2. The KGG Dsrbuon Pascoa e al. 2011) defned he KGG dsrbuon wh fve posve parameers, τ, k, λ and ϕ o exend he GG and EGG dsrbuons poneered by Sacy 1962) and Cordero e al. 2010), respecvely. The KGG probably densy funcon pdf) for > 0) s gven by f) = λ ϕ τ ) τk 1 exp Γk) γ 1 k, ) ] τ ) ]} τ λ 1 γ 1 k, ) ]} τ λ ) ϕ 1, 1) where Γk) = 0 w k 1 e w dw for k > 0) s he gamma funcon, γ 1 k, x) = γk, x)/γk) s he ncomplee gamma funcon rao and γk, x) = x 0 wk 1 e w dw s he ncomplee gamma funcon. The ncomplee gamma funcon defned here

4 784 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. for posve real k and x can be developed no he conex of holomorphc funcons for any complex k and x. Complex analyss shows how properes of he real ncomplee gamma funcons exend o her holomorphc counerpars. In he densy funcon 1), s a scale parameer and τ, k, λ and ϕ are shape parameers. The Webull, GG and EGG dsrbuons are specal models of 1) when λ = k = 1, λ = ϕ = 1 and ϕ = 1, respecvely. The KGG dsrbuon approaches he log-normal dsrbuon when λ = ϕ = 1 and k. Now, we defne an exended form of he densy funcon 1) for > 0) gven by f) = λ ϕ τ ) τk 1 ) τ ) ]} τ λ 1 exp ]γ 1 k, Γk) γ 1 k, ) ]} τ λ ) ϕ 1, 2) where τ s no zero and he oher parameers are posve. The ermnology Kumaraswamy generalzed gamma KGG) dsrbuon s mananed for 2). For a random varable T havng hs densy, we wre T KGG, τ, k, λ, ϕ). For τ > 0, 2) reduces o 1) and for τ > 0 and λ = ϕ = 1, s dencal o he GG dsrbuon. For τ > 0 and λ = ϕ = k = 1, we oban he Webull dsrbuon. The case τ < 0 and λ = ϕ = k = 1 gves he recprocal Webull or nverse Webull) dsrbuon. For τ = 1 and λ = ϕ = k = 1, we oban he recprocal exponenal. If τ = 2 and λ = ϕ = k = 1, we have he recprocal Raylegh. For τ = 1 and λ = ϕ = 1, we oban he recprocal gamma. The case τ < 0 and λ = ϕ = 1 corresponds o he generalzed recprocal gamma. If = 1/2, τ = 1, k = p/2 and λ = ϕ = 1, we have he recprocal ch-square. The values = 1/ 2, τ = 2, k = p/2 and λ = ϕ = 1 yeld he recprocal-ch. If = 2, τ = 2, k = p/2 and λ = ϕ = 1, we oban he scaled recprocal-ch. The case τ < 0 and k = 1 gves he Kumaraswamy Kum for shor) recprocal Webull. For τ = 1 and k = 1, we have he Kum recprocal exponenal. If τ = 2 and k = 1, we oban he Kum recprocal Raylegh. The Kum recprocal gamma corresponds o τ < 0. Several specal models of 2) when τ > 0 are dscussed by Pascoa e al. 2011). Plos of he KGG densy funcon for seleced values of τ > 0 and τ < 0 are dsplayed n Fgure 1. The cumulave dsrbuon funcon cdf) correspondng o 2) becomes γ 1 k, )] } λ ϕ, )τ f τ > 0, F ) = γ 1 k, )] } 3) λ ϕ, )τ f τ < 0.

5 The Log-Kumaraswamy Generalzed Gamma Regresson Model 785 a) b) c) f) =1; τ=0.7; k=1; λ=1; ϕ=1 =1; τ=1.5; k=1; λ=1; ϕ=1 =1; τ=2; k=1; λ=1; ϕ=1 =1; τ=2.5; k=1; λ=1; ϕ=1 =1; τ=3; k=1; λ=1; ϕ=1 f) =1; τ= 0.7; k=1; λ=1; ϕ=1 =1; τ= 1.5; k=1; λ=1; ϕ=1 =1; τ= 2; k=1; λ=1; ϕ=1 =1; τ= 2.5; k=1; λ=1; ϕ=1 =1; τ= 3; k=1; λ=1; ϕ=1 f) =4.7; τ=7.2; k=2.8; λ=0.8; ϕ=1.5 =5; τ=2.3; k=3; λ=0.9; ϕ=2.5 =1.9; τ= 5; k=0.4; λ=2.1; ϕ=2 =1.2; τ= 0.6; k=0.4; λ=10; ϕ=0.8 =6; τ=3; k=1; λ=20; ϕ=0.9 =3.1; τ= 10.3; k=2; λ=0.1; ϕ= Fgure 1: The KGG densy curves: a) For some values of τ > 0. b) For some values of τ < 0. c) For some values of τ > 0 and τ < 0 Le g,τ,k ) be he GG, τ, k) densy funcon Sacy and Mhram, 1965) gven by for > 0) g,τ,k ) = τ ) τk 1 ) ] τ exp. 4) Γk) The generalzed bnomal expanson converges for z < 1 and any real b ) b z) b = 1) j z j, 5) j j=0 where b j) = Γb + 1)/Γb j + 1)j!]. Then, from 5), he densy funcon 2) can be expressed as f) = λ ϕ τ Γk) γ 1 ) τk 1 ) ] τ ) ]} τ λ 1 ) ϕ 1 exp γ 1 k, 1) j j j=0 k, ) ]} τ λj. Ths equaon holds for any ϕ > 0. From expanson 27) gven n Appendx A), we oban f) = λ ϕ τ ) τk 1 ) ] τ ) ϕ 1 exp 1) j s m λ 1) Γk) j m,j=0 ) ]} τ λj+m γ 1 k,. 6)

6 786 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. For any real λj + m > 0, we can wre ) ]} τ jλ+m ) γ λj + m 1 k, = 1) l γ 1 l l=0 k, ) ]} τ l. Applyng he bnomal expanson, 6) can be rewren as f) = λ ϕ τ ) kτ 1 ) ] τ l exp 1) j+l+q s m λ 1) Γk) j,l,m=0 q=0 ) ) ) ) ]} ϕ 1 jλ + m l τ q γ 1 k,. j l q From 31) gven n Appendx A) and replacng l l=0 q=0 by q=0 l=q, he densy funcon f) can be expressed as a double lnear combnaon of GG densy funcons f) = d,q=0 w d,q g,τ,k1+q)+d ), > 0, 7) where g,τ,k1+q)+d ) s he GG, τ, k1 + q) + d) densy funcon gven by 4) and he coeffcens w d,q are w d,q = w d,q k, λ, ϕ) = λ ϕ ) ϕ 1 λj + m j l j,m=0 l=q ) l q 1) j+l+q Γk1 + q) + d] s m λ 1) c q,d Γk) q+1 ), 8) where he quany s m λ 1) s gven by 28) and he quanes c q,d can be deermned from he recurrence 30) see Appendx A). Clearly, d,q=0 w d,q = 1. 7) s he man resul of hs secon. Some KGG mahemacal properes for example, he ordnary, nverse, facoral and ncomplee momens and generang funcon can be obaned from 7) and hose GG properes. 3. General Properes Henceforh, le T and Y be random varables havng he denses 2) and 4), respecvely. 3.1 Momens Here, we oban an nfne represenaon for he rh ordnary momen of T,

7 The Log-Kumaraswamy Generalzed Gamma Regresson Model 787 say µ r = ET r ). The rh momen of he GG, τ, k) dsrbuon s Based on 7), µ r can be expressed as µ r,gg = r Γk + r/τ). Γk) µ r = r 1) j+l+q Γk1 + q) + d + r/τ] s m λ 1) c q,d λ ϕ Γk) q+1 d,q,j,m=0 l=q ) ) ) ϕ 1 λj + m l, j l q where he quany s m λ) s gven by 28) and he coeffcens c q,d can be deermned from 30). 3.2 Generang Funcon Here, we provde an explc expresson for he mgf of Y, say M,τ,k s) = Ee sy ), for τ > 1, based on he Wrgh funcon Wrgh, 1935). We are unable o oban a closed-form expresson for M,τ,k s) when τ < 0. We can wre M,τ,k s) = Seng u = /, we have M,τ,k s) = τ τk Γk) τ Γk) 0 0 exps) τk 1 exp /) τ }d. expsu) u τk 1 exp u τ )du. Now, we assume τ > 0. Expandng he frs exponenal n Taylor seres and usng 0 u kτ+m 1 e uτ du = τ 1 Γm/τ + k), we oban M,τ,k s) = sgnτ) Γk) m ) s) m Γ τ + k. 9) m! m=0 9) holds for any real τ > 0. However, for τ > 1, can be smplfed by consderng he Wrgh generalzed hypergeomerc funcon Wrgh, 1935) defned by ) ) ] 1, A 1,, p, A p pψ q ) β1, B 1,, βq, B q ) ; x = m=0 p Γ j + A j m) j=1 q Γβ j + B j m) j=1 x m m!.

8 788 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. Ths funcon exss f 1 + q j=1 B j p j=1 A j > 0. By combnng he las wo equaons, we have M,τ,k s) = 1 k, τ 1 ) Γk) 1 Ψ 0 ] ; s. 10) Fnally, he mgf of T for τ > 1) can be wren from 7) and 10) as Ms) = d,q=0 w d,q k1 + q] + d, τ 1 ) Γk1 + q] + d) 1 Ψ 0 9)-11) are he man resul of hs secon. 3.3 Order Sascs ] ; s. 11) The densy funcon f :n ) of he h order sasc, say T :n, for = 1,, n, from random varables T 1,, T n havng densy 2), s gven by f :n ) = 1 B, n + 1) f) F ) 1 F )} n, where f) and F ) are he pdf and cdf of he KGG dsrbuon, respecvely and Bp, q) = Γp)Γq)/Γp + q) denoes he bea funcon. We readly oban usng he bnomal expanson f :n ) = n 1 B, n + 1) f) ) 1) j 1 n F ) +j1 1. j 1 =0 j 1 Now, we derve an expresson for he densy funcon of he KGG order sascs n erms of he baselne densy funcon mulpled by a power seres of G) = γ 1 k, /) τ ). Seng ϕ = λ = 1 n 3), s clear ha he cdf of he GG, τ, k) dsrbuon s G) = γ 1 k, /) τ ) for τ > 0 and G) = γ 1 k, /) τ ) for τ < 0. Ths resul enables us o oban he ordnary momens of he KGG order sascs as nfne weghed sums of convenen quanes defned by δ s,r = 0 ) τ ] r s γ 1 k, g,τ,k)d. These quanes δ s,r can be easly negraed numercally. They represen he probably weghed momens PWMs) of Y when τ > 0. We shall consder wo dsnc cases: τ > 0 and τ < 0.

9 For τ > 0 The Log-Kumaraswamy Generalzed Gamma Regresson Model 789 An expanson for F ) +j 1 1 follows from 3) as F ) +j 1 1 = +j 1 l 1 =0 1) l 1 ) + j 1 G) λ] l 1 ϕ. Usng he seres expanson for G) λ] l 1 ϕ, we oban F ) +j 1 1 = +j 1 l 1 =0 l 1 l 1 ) 1) l 1 + j 1 ) 1) m 1 l1 ϕ G) m1λ. m 1 =0 For m 1 λ > 0, we can wre G) m1λ = r=0 s rm 1 λ) G) r, where he coeffcens s r m 1 λ) can be obaned from 28) as s r m 1 λ) = for r = 0, 1,. Thus, F ) +j 1 1 = +j 1 l 1 =0 l 2 =r 1) l 2+r m1 λ l 2 ) ) l2, r m 1 ) 1) l 1 + j 1 υ r l 1, λ, ϕ) G) r, l 1 r=0 where υ r l 1, λ, ϕ) = ) 1) m 1 l1 ϕ s r m 1 λ). m 1 =0 m 1 Inerchangng he sums, we oban where p r,u λ, ϕ) = F ) +j 1 1 = p r,+j1λ, ϕ) G) r, r=0 u ) 1) l 1 u l 1 =0 l 1 m 1 =0 l 2 =r ) 1) m 1+r+l 2 l1 ϕ m1 λ m 1 for r, u = 0, 1,. The densy funcon of T :n becomes l 2 ) l2 r ), f :n ) = n 1 B, n + 1) f) ) 1) j 1 n p r,+j1λ, ϕ) G) r, j 1 =0 j 1 r=0

10 790 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. and hen 1 n ) f :n ) = 1) j 1 n w d,q p r,+j1λ, ϕ) γ 1 B, n + 1) j d,q,r=0 j 1 =0 1 ) τ ] r k, g,τ,k1+q)+d). The las equaon can be rewren as where f :n ) = n d,q,r=0 j 1 =0 k, md, q,, j 1, n) p r,+j1λ, ϕ) τkq+d) γ 1 ) τ ] r g,τ,k), 12) λ ϕ 1) j+j1+l+q s m λ 1) c q,d md, q,, j 1, n) = B, n + 1) τkq+d) Γk) q j,m=0 l=q ) ) ) ) ϕ 1 n λj + m l. 13) j l q Then, for τ > 0, E T:n s ) can be expressed as E T s :n) = n d,q,r=0 j 1 =0 where δ s+τkq+d),r was defned before. For τ < 0 From 3), we can wre j 1 md, q,, j 1, n) p r,+j1λ, ϕ) δ s+τkq+d),r, F ) +j1 1 = G) λ] ϕ+j 1) ) = 1) m 1 ϕ + j 1) G) m1λ. m 1 =0 For m 1 λ > 0, we have G) m 1λ = r=0 s rm 1 λ) G) r, where s r m 1 λ) s gven by 28) for r = 0, 1,. Furher, F ) +j 1 1 = q r, j 1, λ, ϕ) G) r, r=0 m 1

11 where The Log-Kumaraswamy Generalzed Gamma Regresson Model 791 q r, j 1, λ, ϕ) = ) 1) m 1 ϕ + j 1) s r m 1 λ). m 1 =0 The densy funcon of he KGG order sascs becomes m 1 f :n ) = n 1 B, n + 1) f) ) 1) j 1 n q r, j 1, λ, ϕ) G) r, j 1 =0 j 1 r=0 and hen 1 n ) f :n ) = 1) j 1 n w d,q q r, j 1, λ, ϕ)γ 1 B, n + 1) j d,q,r=0 j 1 =0 1 ) τ ] r k, g,τ,k1+q)+d). The las equaon can be rewren as f :n ) = n d,q,r=0 j 1 =0 k, md, q,, j 1, n) τkq+d) q r, j 1, λ, ϕ)γ 1 ) τ ] r g,τ,k), 14) where md, q,, j 1, n) s gven by 13). So, he momens of he KGG order sascs for τ < 0 can be expressed as E T s :n) = n d,q,r=0 j 1 =0 12) and 14) are he man resuls of hs secon. 4. The LKGG Dsrbuon md, q,, j 1, n)q r, j 1, λ, ϕ) δ s+τkq+d),r. Henceforh, le T be a random varable followng he KGG densy funcon 2) and le Y = logt ). Seng k = q 2, τ = k) 1 and = exp µ τ 1 logk) ], he densy funcon of Y can be expressed as fy) = λ ϕ q q 2 ) q 2 Γq 2 exp q 1 y µ ) y µ γ 1 q 2, q 2 exp q )}]} λ 1 ) q 2 exp q y µ )]} γ 1 q 2, q 2 exp y µ )})] λ } ϕ 1, q

12 792 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. where < y, µ <, > 0, λ > 0, ϕ > 0 and q s dfferen from zero. We consder an exended form ncludng he case q = 0 Lawless, 2003). Thus, he densy of Y can be expressed as ) )]} λ ϕ q q 2 ) q 2 exp q 1 y µ Γq 2 ) q 2 exp q y µ )}]} λ 1 γ 1 q 2, q 2 exp fy) = q γ 1 q 2, q 2 exp λ ϕ 2π exp 1 2 y µ y µ q y µ ) 2 }Φ λ 1) y µ )})] λ } ϕ 1, f q 0, ) Φ λ y µ )] ϕ 1, f q = 0, where Φ ) s he sandard normal cumulave dsrbuon. We refer o 15) as he log-kumaraswamy generalzed gamma LKGG) dsrbuon, say Y LKGGµ,, q, λ, ϕ), where µ R s he locaon parameer, > 0 s he scale parameer and q, λ and ϕ are shape parameers. For q = 0 and λϕ = 1 and q = 1, we have from 15) he skew normal and exreme value dsrbuons, respecvely. For ϕ = 1 and λ = ϕ = 1, we oban he log-exponenaed generalzed gamma Orega e al., 2012) and log-gamma generalzed dsrbuons, respecvely. The case λ = ϕ = 1 and q = 1 corresponds o he log-nverse Webull dsrbuon. Thus, f T KGG, τ, k, λ, ϕ) hen Y = logt ) LKGGµ,, q, λ, ϕ). Seng µ = 0 and = 1, he plos of he densy funcon 15) for seleced values of λ and ϕ, when q < 0, q > 0 and q = 0, are dsplayed n Fgure 2. These plos clearly ndcae ha he LKGG dsrbuon could be very flexble for modelng s kuross. 15) fy) q=0.4; λ=0.6; ϕ=0.9 q=4; λ=3; ϕ=0.5 q=0.2; λ=0.4; ϕ=1.2 q=0.5; λ=1.9; ϕ=0.8 q=0.1; λ=0.7; ϕ=5 fy) q= 0.4; λ=0.6; ϕ=0.9 q= 4; λ=3; ϕ=0.5 q= 0.2; λ=0.4; ϕ=1.2 q= 0.5; λ=1.9; ϕ=0.8 q= 0.1; λ=0.7; ϕ=5 fy) λ=0.6; ϕ=0.9 λ=3; ϕ=0.5 λ=0.4; ϕ=1.2 λ=1.9; ϕ=0.8 λ=0.7; ϕ= y Fgure 2: The LKGG densy curves: a) For some values of q > 0. b) For some values of q < 0. c) For some values of q = 0 y y

13 The Log-Kumaraswamy Generalzed Gamma Regresson Model 793 For q > 0, he survval funcon of Y s gven by Sy) = F y) = P Y > y) = P µ + Z > y) = P Z > z) γ1 = λϕqq 2 ) q 2 exp q 1 u q 2 expqu)} q 2, q 2 expqu) ]} λ 1 du Γq 2 ) γ1 q 2, q 2 expqu) ]} λ] ϕ 1) ; z For q < 0, he survval funcon of Y s gven by Sy) = F y) = P Y > y) = P µ + Z > y) = P Z > z) γ1 = λϕqq 2 ) q 2 exp q 1 u q 2 expqu)} q 2, q 2 expqu) ]} λ 1 du Γq 2 ) γ1 q 2, q 2 expqu) ]} λ] ϕ 1). z These negrals can be reduced o γ 1 q 2, q 2 exp } q y µ )])] λ ϕ, f q > 0, Sy) = γ 1 q 2, q 2 exp } q y µ )])] λ ϕ, f q < 0, 16) Φ λ y µ ) ] ϕ, f q = 0. Now, we derve he rh momen of Y LKGGµ,, q, λ, ϕ), say µ r. For q 0, we oban µ r = y r λϕ q q 2) q 2 Γq 2 exp q 2 y µ ) y µ )]}) q exp q ) y µ )]}) λ 1 γ 1 q 2, q 2 exp q y µ )]}) λ ] ϕ 1dy. γ 1 q 2, q 2 exp q Seng x = q 2 exp q y µ )] and usng 5), µ r reduces o µ λϕ sgnq) } r r = Γ q 2 logx) + 2 log q )] + µ x q 2 1 e x ) 0 q ) ϕ 1 γ1 1) j q 2, x )] λj+1) 1 dx. j j=0

14 794 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. Applyng expanson 27) n Appendx A for γ 1 q 2, x )] λj+1) 1, we oban } µ λϕ sgnq) r r = 0 Γ q 2 logx) + 2 log q )] + µ x q 2 1 e x ) q ) ϕ 1 1) j s m λj + 1) 1) γ 1 q 2, x ) m dx. j j,m=0 Usng expanson 31) n Appendx A for γ 1 q 2, x ) m, we have } µ λϕ sgnq) r r = 0 Γ q 2 logx) + 2 log q )] + µ e x ) q 1) j ) s m λj + 1) 1) c m, ϕ 1 Γq 2 ) m x q 2 m+1)+ 1 dx. j j,m,=0 Noe ha 2 q log q ) + µ + } r q logx) = r ) ] r 2 r l ) l l q log q ) + µ l logx), q l=0 and hen µ λϕ sgnq) r 1) j ) ) s m λj + 1) 1) c m, ϕ 1 r r = Γq 2 ) Γ q 2 ) m j l j,m,=0 l=0 ] 2 r l ) l q log q ) + µ l logx)e x x q 2 m+1)+ 1 dx. q The las negral s gven n Prudnkov e al. 1986, Vol 1, Secon ) and hen µ λϕ sgnq) r 1) j ) ) s m λj + 1) 1) c m, ϕ 1 r r = Γq 2 ) Γ q 2 ) m j l j,m,=0 l=0 2 q log q ) + µ where Γp) = Γp)/ p. For q = 0, we can wre µ r = EY r ) = λϕ 2π Φ λ) y µ 0 ] r l Γq 2 m + 1) + )] q y r exp 1 2 )] ϕ 1dy. y µ ) 2 ] ) ll, Φ λ 1) y µ )

15 The Log-Kumaraswamy Generalzed Gamma Regresson Model 795 Usng 5), we have µ r = λϕ 2π y µ ) dy. y r exp 1 2 y µ ) 2 ] ) ϕ 1 1) m m m=0 Φ λm+1) 1) The PWM for j and p non-negave negers) of he sandardzed normal dsrbuon s ν j,p = z j φz) Φ p z)dz, where φz) s he sandard normal densy. Seng y = µ + z and usng 27) n Appendx A for Φ λm+1) 1) z), we oban µ r = λ ϕ r m,p=0 j=0 1) m ϕ 1 m ) r j ) s p λm + 1) 1) j µ r j ν j,p, 17) The sandard cumulave normal can be expressed as Φx) = 1 )} x 1 + erf, x R. 2 2 Usng he bnomal expanson and nerchangng erms, we oban ν j,p = 1 2 p 2π p l=0 ) p ) x p l x j e x2 /2 erf dx. l 2 From he power seres for he error funcon erf ) erfx) = 2 π m=0 1) m x 2m+1 2m + 1) m!, he las negral follows from 9)-11) of Nadarajah 2008b). When j +p l s even, we have ν j,p = 2 j/2 π p+1)/2 F p l) A p l=0 j+p l) even j + p l ) p 2 l π l/2 Γ l ) j + p l + 1 ; 1 2,, 1 2 ; 3 2,, 3 2 ; 1,, 1 ), 18) 2

16 796 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. where F p l) A s gven n erms of he Laurcella funcon of ype A Exon, 1978; Aars, 2000) defned by F n) A a; b 1,, b n ; c 1,, c n ; x 1,, x n ) a) m1 + +m n b 1 ) = m1 b n ) mn x m1 1 x mn n c 1 ) m1 c n ) mn m 1! m n!, m 1 =0 m n=0 and a) = aa + 1) a + 1) s he ascendng facoral wh he convenon ha a) 0 = 1. Numercal rounes for he drec compuaon of he Laurcella funcon of ype A are avalable, see Exon 1978) and Mahemaca Tro, 2006). Plos of he skewness and kuross for seleced values of µ,, q, λ and ϕ are dsplayed n Fgures 3 and 4, respecvely. Skewness ϕ=0.5 ϕ=1 ϕ=1.5 ϕ=2 Kuross ϕ=0.5 ϕ=1 ϕ=1.5 ϕ= λ Fgure 3: Skewness and kuross of he LKGG dsrbuon as a funcon of λ for some values of ϕ wh µ = 1.5, = 2 and q = 0.5 λ Skewness λ=0.5 λ=1 λ=1.5 λ=2 Kuross λ=0.5 λ=1 λ=1.5 λ= ϕ Fgure 4: Skewness and kuross of he LKGG dsrbuon as a funcon of ϕ for some values of λ wh µ = 1.5, = 2 and q = 0.5 ϕ

17 The Log-Kumaraswamy Generalzed Gamma Regresson Model The LKGG Regresson Model In many praccal applcaons, he lfemes are affeced by explanaory varables such as he choleserol level, blood pressure, wegh and many ohers. Paramerc models o esmae unvarae survval funcons for censored daa regresson problems are wdely used. A paramerc model ha provdes a good f o lfeme daa ends o yeld more precse esmaes of he quanes of neres. If Y LKGGµ,, q, λ, ϕ), we defne he sandardzed random varable Z = Y µ)/ havng densy funcon gven by fz) = λ ϕ q Γq 2 ) q 2 ) q 2 exp q 1 z q 2 expqz) } γ 1 q 2, q 2 expqz) ]} λ 1 γ 1 q 2, q 2 expqz) ]) λ} ϕ 1, f q 0, λ ϕ 2π exp ) z2 Φ λ 1) z) Φ λ z)] ϕ 1, f q = 0. 2 We wre Z LKGG0, 1, q, λ, ϕ). Furher, we propose a lnear locaon-scale regresson model lnkng he response varable y and he explanaory varable vecor x T = x 1,, x p ) by 19) y = x T β + z, = 1,, n, 20) where he random error z has densy funcon 19), β = β 1,, β p ) T, > 0, λ > 0 and < q < are unknown parameers. The parameer µ = x T β s he locaon of y. The locaon parameer vecor µ = µ 1,, µ n ) T s defned by a lnear model µ = Xβ, where X = x 1,, x n ) T s a known model marx. The LKGG model 20) opens new possbles for fed many dfferen ypes of daa. I conans as specal models some well-known regresson models. For ϕ = 1, we oban he log-exponenaed generalzed gamma LEGG) regresson model Orega e al., 2012). The case λ = 1 leads o he log-exponenaed complemen generalzed gamma LECGG) regresson model, whereas ϕ = λ = 1 yelds he log-gamma generalzed LGG) regresson model. For ϕ = λ = q = 1, we oban he classcal log-webull or exreme value) regresson model see, Lawless, 2003). If = 1 and = 0.5, n addon o ϕ = λ = q = 1, he new regresson model reduces o he exponenal and Raylegh regresson models, respecvely. The case λ = ϕ = q = 1 refers o he log-exponenaed Webull LEW) regresson model Mudholkar e al., 1995). See, also, Cancho e al. 1999, 2009), Orega e al. 2006) and Hashmoo e al. 2010). If = 1, n addon o q = 1, concdes wh he log-exponenaed exponenal regresson model. If = 0.5, n addon o q = 1, gves he log-generalzed Raylegh regresson model. For λ = 1, we have he log-gamma generalzed LGG) regresson model Lawless, 2003). Recenly, he LGG dsrbuon has been used n several research areas.

18 798 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. See, for example, Orega e al. 2003, 2008, 2009). For q = 1, we oban he log-generalzed nverse Webull regresson model. If λ = 1, n addon o q = 1, follows he log-nverse Webull regresson model. For q = 0, we have he logexponenaed normal regresson model. Fnally, for λ = 2, he LKGG regresson model reduces o he skew normal regresson model. 5.1 Maxmum Lkelhood Esmaon Consder a sample y 1, x 1 ),, y n, x n ) of n ndependen observaons, where each random response s defned by y = mnlog ), logc )}. We assume nonnformave censorng such ha he observed lfemes and censorng mes are ndependen. Le F and C be he ses of ndvduals for whch y s he log-lfeme or log-censorng, respecvely. Convenonal lkelhood esmaon echnques can be appled here. The log-lkelhood funcon for he vecor of parameers θ = q, λ, ϕ,, β T ) T from model 20) has he form lθ) = l θ) + C l c) θ), where l θ) = logfy )], l c) θ) = logsy )], fy ) s he densy funcon 15) and Sy ) s he survval funcon 16) of Y. The oal log-lkelhood funcon for θ can be paroned as lθ) = ] v log λ ϕ q q 2 ) q 2 + q 1 z Γq 2 ) q 2 expqz ) +λ 1) logγ 1 q 2, q 2 expqz )]} +ϕ 1) log γ 1 q 2, q 2 expqz )]} λ} +ϕ log γ 1 q 2, q 2 expqz )]} λ}, f q > 0, C ] v log λ ϕ q) q 2 ) q 2 + q 1 z Γq 2 ) q 2 expqz ) +λ 1) logγ 1 q 2, q 2 expqz )]} +ϕ 1) log γ 1 q 2, q 2 expqz )]} λ} + log γ 1 q 2, q 2 expqz )]} λ] ϕ}, f q < 0, C ] v log 1 2 z 2 + λ 1) logφz )] λϕ 2π +ϕ 1) log Φ λ z )] + ϕ C log Φ λ z )], f q = 0, 21) where v s he number of uncensored observaons falures) and z = y x T β)/. The maxmum lkelhood esmae MLE) θ of he vecor of he model parameers can be calculaed by maxmzng he log-lkelhood 21). Inal values for β and are obaned by fng he Webull regresson model wh λ = 1, ϕ = 1 and

19 The Log-Kumaraswamy Generalzed Gamma Regresson Model 799 q = 1. If ẑ = y x T ˆβ)/ˆ, he f of he LKGG model yelds he esmaed survval funcon for y Ŝy ; ˆλ, ˆϕ, ˆ, ˆq, β T ) = γ1 ˆq 2, ˆq 2 expˆqẑ )]}ˆλ } ˆϕ, f q > 0, γ 1 ˆq 2, ˆq 2 ) ˆϕ, expˆqẑ ]}ˆλ} f q < 0, 22) Φˆλẑ ) ] ˆϕ, f q = 0. Under condons ha are fulflled for he parameer vecor θ n he neror of he parameer space bu no on he boundary, he asympoc dsrbuon of n θ θ) s mulvarae normal Np+3 0, Kθ) 1 ), where Kθ) s he nformaon marx. The asympoc covarance marx Kθ) 1 of θ can be approxmaed by he nverse of he p + 3) p + 3) observed nformaon marx Lθ). The elemens of Lθ), namely L λλ, L λϕ, L λ, L λβj, L ϕϕ, L ϕ, L ϕβj, L, L βj and L βj β s for j, s = 1,, p, are gven n Appendx B. The approxmae mulvarae normal dsrbuon N p+3 0, Lθ) 1 ) for θ can be used n he classcal way o consruc approxmae confdence regons for some componens of θ. Furher, we can use he lkelhood rao LR) sasc for comparng some sub-models wh he LKGG model. We consder he paron θ = θ T 1, θ T 2 ) T, where θ 1 s a subse of parameers of neres and θ 2 s a subse of remanng parameers. The LR sasc for esng he null hypohess H 0 : θ 1 = θ 0) 1 versus he alernave hypohess H 1 : θ 1 θ 0) 1 s gven by w = 2l θ) l θ)}, where θ and θ are he esmaes under he null and alernave hypoheses, respecvely. The sasc w s asympocally as n ) dsrbued as χ 2 k, where k s he dmenson of he subse of parameers θ 1 of neres. 5.2 A Bayesan Analyss As an alernave analyss, we use he Bayesan mehod whch allows for he ncorporaon of prevous knowledge of he parameers hrough nformave pror densy funcons. When hs nformaon s no avalable, we can consder a nonnformave pror. In he Bayesan approach, he nformaon referrng o he model parameers s obaned hrough a poseror margnal dsrbuon. In hs way, wo dffcules usually arse. The frs refers o aanng margnal poseror dsrbuon, and he second o he calculaon of he momens of neres. Boh cases requre numercal negraon ha, many mes, do no presen an analycal soluon. Here, we use he smulaon mehod of Markov Chan Mone Carlo MCMC), such as he Gbbs sampler and Meropols-Hasngs algorhm.

20 800 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. Snce we have no pror nformaon from hsorcal daa or from prevous expermen, we assgn conjugae bu weakly nformave pror dsrbuons o he parameers. Snce we assume nformave bu weakly) pror dsrbuon, he poseror dsrbuon s a well-defned proper dsrbuon. Here, we consder ha he elemens of he parameer vecor are ndependen and ha he jon pror dsrbuon of all unknown parameers has densy funcon gven by πλ, ϕ,, q, β) πλ) πϕ) π) πq) πβ). 23) Here, λ Γa 1, b 1 ), ϕ Γa 2, b 2 ), Γa 3, b 3 ), q N µ 1, 1 2) and β s N µ s, s), 2 where Γa, b ) denoes a gamma dsrbuon wh mean a /b, varance a /b 2 and densy funcon fυ; a, b ) = ba υa 1 exp υb ), Γa ) where υ > 0, a > 0 and b > 0, N µ s, 2 s) denoes he normal dsrbuon wh mean µ s, varance 2 s and densy funcon gven by fx; µ s, s ) = 1 2π 2 s exp x µ s) 2 where x, µ s R and s 2 > 0. All hyper-parameers are specfed. Combnng he lkelhood funcon 21) and he pror dsrbuon 23), he jon poseror dsrbuon for λ, ϕ,, q and βs T reduces o ] υ λϕ q)q 2 ) q 2 πλ, ϕ,, q, β y) exp Γq 2 q ] 1 y x T β) ) exp q ]}) 2 y x T exp q β) ]})] λ 1 γ 1 q 2, q 2 y x T exp q β) ]})] λ } ϕ 1 γ 1 q 2, q 2 y x T exp q β) ]})] λ } ϕ γ 1 q 2, q 2 y x T exp q β) C πλ, ϕ,, q, β). 2 2 s ],

21 The Log-Kumaraswamy Generalzed Gamma Regresson Model 801 The jon poseror densy above s analycally nracable because he negraon of he jon poseror densy s no easy o perform. So, he nference can be based on MCMC smulaon mehods such as he Gbbs sampler and Meropols-Hasngs algorhm, whch can be used o draw samples, from whch feaures of he margnal dsrbuons of neres can be nferred. In hs drecon, we frs oban he full condonal dsrbuons of he unknown parameers gven by πλ y, ϕ,, q, β) λ ]})] λ γ υ 1 q 2, q 2 y x T exp q β) ]})] λ } ϕ 1 γ 1 q 2, q 2 y x T exp q β) ]})] λ } ϕ γ 1 q 2, q 2 y x T exp q β) C πλ), πϕ y, λ,, q, β) ϕ υ π y, λ, ϕ, q, β) υ exp γ 1 C C πϕ), q 1 q 2, q 2 exp γ 1 γ 1 q 2, q 2 exp γ 1 q q 2, q 2 exp ]})] λ } ϕ y x T β) q ]})] λ } ϕ y x T β) ] y x T β) exp q ]}) 2 y x T exp q β) q q 2, q 2 exp γ 1 ]})] λ 1 y x T β) q q 2, q 2 exp ]})] λ } ϕ 1 y x T β) q ]})] λ } ϕ y x T β) π),

22 802 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. ] υ q)q 2 ) q 2 πq y, λ, ϕ,, β) exp Γq 2 q ] 1 y x T β) ) exp q ]}) 2 y x T exp q β) ]})] λ 1 γ 1 q 2, q 2 y x T exp q β) ]})] λ } ϕ 1 γ 1 q 2, q 2 y x T exp q β) ]})] λ } ϕ γ 1 q 2, q 2 y x T exp q β) C πq) and πβ y, λ, ϕ,, q) exp q 1 ] ]}) y x T β) exp q 2 y x T exp q β) γ 1 C πβ). q 2, q 2 exp γ 1 q q 2, q 2 exp γ 1 ]})] λ 1 y x T β) q q 2, q 2 exp ]})] λ } ϕ 1 y x T β) q ]})] λ } ϕ y x T β) Snce he full condonal dsrbuons for λ, ϕ,, q and β do no have closedform, we requre he use of he Meropols-Hasngs algorhm. The jon poseror densy for q > 0 s defned analogously, consderng he lkelhood funcon 21) for q > 0 and he pror dsrbuon 23). Combnng he lkelhood funcon 21) and he pror dsrbuon 23), he jon poseror dsrbuon for λ, ϕ, and βs T for q = 0 reduces o

23 The Log-Kumaraswamy Generalzed Gamma Regresson Model 803 ) υ ) 2 ] )] λ 1 λϕ πλ, ϕ,, β y) exp 1 y x T β y x T Φ β 2π 2 )] λ } ϕ 1 )] λ } ϕ y x T Φ β y x T Φ β C πλ, ϕ,, β). The condonal dsrbuons for hese parameers akng q = 0 are gven by πλ y, ϕ,, β) λ )] λ )] λ } ϕ 1 υ y x T Φ β y x T Φ β )] λ } ϕ y x T Φ β πλ), C πϕ y, λ,, β) ϕ υ πϕ), Φ y x T β )] λ } ϕ Φ C )] λ } ϕ y x T β ) 2 ] )] λ 1 π y, λ, ϕ, β) exp 1 y x T β y x T Φ β 2 )] λ } ϕ 1 )] λ } ϕ y x T Φ β y x T Φ β C πλ), and ] )] λ 1 πβ y, λ, ϕ, ) exp 1 x T β 2y ) y x T 2 2 Φ β )] λ } ϕ 1 )] λ } ϕ y x T Φ β y x T Φ β C πβ). The MCMC compuaons were mplemened n he sascal sofware package R.

24 804 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. 6. Applcaon: Chemcal Dependency Daa We demonsrae he poenaly of he LKGG regresson model appled o chemcal dependency paens as nvesgaed by Pascoa 2008). The daa were provded by he Assocaon Moher Brave, localzed n he cy of Caranga, MG, Brazl, where n = 141 resdens were consdered drug addcs from 2000 o The response varable was he me spen n he communy unl he whdraw of he reamen, consderng ha each resden remans n he communy for a maxmum perod of 270 days whou any conac wh drugs or free drugs. The resden who acheves hs goal was regarded as a censored observaon. Chemcal dependency s a chronc dsease of he bran Kalvas and Volkow, 2005). One of s man characerscs s he relapse phenomenon, whereby sufferers reurn o her consumpon of he problem subsance, followed by a renewed aemp o sop or reduce hs consumpon Brandon e al., 2007; Koob and Le Moal, 1997). Relapse does no mean falure of reamen. Raher s a par of he rehablaon process Marla, 2001). The paen s n relapse when he or she presens all or par of he dysfunconal behavor paerns ha were presen before reamen. Thus, relapse sars before acual renewed consumpon. I s marked by he occurrence of faclang smul wh cognve, emoonal, physcal and socal aspecs Maso e al., 2003). Behavoral changes and exposure o hgh-rsk suaons generally precede resumed consumpon of he subsance Mller e al., 1996). Even he mos movaed paens can presen relapse epsodes Baker e al., 2004). Neverheless, relapse s predcable and avodable Brandon e al., 2007). As he me of absnence ncreases, he chance of relapse decreases, alhough hs rsk s never oally elmnaed durng he lfeme of a person wh an addcon. Sudes of he rsk facors assocaed wh relapse n chemcal dependency are very mporan, because her fndngs can ncrease he probably of suable early clncal nervenons. Furhermore, rsk sudes allow denfcaon of proecve facors o reduce vulnerably and favor ressance o he empaon o relapse. The varables nvolved n he sudy are: : he me spen n he communy unl he whdrawal of reamen days); δ : he censorng ndcaor 0 = censorng, 1 = lfeme observed); x 1 : he maral saus 0 = sngle, 1 = marred); x 2 : schoolng 0 = no educaon or ncomplee bases, 1 = elemenary educaon and hgher educaon); x 3 : age 0 =< 30 years, 1 = 30 years).

25 The Log-Kumaraswamy Generalzed Gamma Regresson Model 805 We f he LKGG regresson model y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + z, where he errors z 1,, z 141 are ndependen random varables havng densy funcon 19). Table 1 lss he MLEs of he parameers for he LKGG regresson model fed o he curren daa usng he NLMxed procedure n SAS. Ths model nvolves an exra parameer whch gves more flexbly o f hese daa. Inal values for β, q and are aken from he f of he LGG regresson model wh λ = 1 and ϕ = 1. The explanaory varables x 1, x 2 and x 3 are margnally sgnfcan for he LKGG model a he sgnfcance level of 5%. Table 1: MLEs of he parameers, sandard errors, p-values and 95% confdence nervals for he LKGG model fed o he chemcal dependency daa Parameer Esmae SE p-value CI 95% q ; ) λ ; ) ϕ ; ) ; ) β < ; ) β < ; ) β < ; ) β < ; ) Cox 1972) proposed a very useful regresson model for analyzng censorng falure mes, where he random varable of neres represens falure me and he falures mes are assumed dencally dsrbued n some specfed form. He noed ha f he proporonal hazards assumpon holds or, s assumed o hold) hen s possble o esmae he effec parameers) whou any consderaon of he hazard funcon non-paramerc approach). Ths approach o survval daa s called proporonal hazards model. The Cox model may be specalzed f a reason exss o assume ha he baselne hazard follows a paramerc form. In hs case, he baselne hazard can be replaced by a paramerc densy. Typcally, we can hen maxmze he full lkelhood whch grealy smplfes model-fng and provdes nerpreably a he cos of flexbly. Le R ) be he se of ndvduals a rsk a me. Condonally on he rsk ses, he requred lkelhood Lβ) can be expressed as Lβ) = n =1 ] expx T β) δ expx T β), 25) j R )

26 806 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. where δ s he censorng ndcaor. The MLE ˆβ of β can be calculaed by maxmzng he lkelhood funcon 25) usng he R sofware. In Table 2, we ls he esmaes, correspondng sandard errors and p-values for he fed Cox regresson model. Explanaory varables x 2 and x 3 are margnally sgnfcan a he 5% sgnfcance level. Table 2: Esmaes of he Cox model fed o he daa of chemcal dependency and correspondng hazard raos HR j ) covarae Esmae SE p-value HR j CI 95% HR j ) β ; ) β ; ) β ; ) Furher, Table 3 gves he Akake Informaon Creron AIC), Conssen Akake Informaon Creron CAIC) and Bayesan Informaon Creron BIC) o compare he LKGG model and Cox proporonal hazard regresson models. The LKGG regresson model ouperforms he oher models rrespecve of he crera and can be used effecvely n he analyss of hese daa. So, he proposed model s a grea alernave o model survval daa. Table 3: The AIC, CAIC and BIC sascs for comparng he LKGG model and Cox proporonal hazard regresson models Model AIC CAIC BIC LKGG Cox In order o assess f he model s approprae, we f he LKGG regresson model for each explanaory varable. In Fgures 5a, b, c, d, we plo he emprcal survval funcon and he esmaed survval funcon 22) for each explanaory varable. We can conclude ha he LKGG regresson model provdes a good f o hese daa. Bayesan analyss The followng ndependen prors were consdered o perform he Meropols- Hasngs algorhm: λ Γ0.01, 0.01), ϕ Γ0.01, 0.01), Γ0.01, 0.01), q N 0, 10) and β s N 0, 10), so ha we have a vague pror dsrbuon. Consderng hese pror densy funcons, we generae wo parallel ndependen runs of he Meropols-Hasngs wh sze 100, 000 for each parameer, dsregardng he frs 10, 000 eraons o elmnae he effec of he nal values and, o avod correlaon problems, we consder a spacng of sze 10, obanng a sample of sze 9, 000 from each chan. To monor he convergence of he Meropols-Hasngs,

27 The Log-Kumaraswamy Generalzed Gamma Regresson Model 807 Sy;x) Kaplan Meer x1=0) Kaplan Meer x1=1) LKGG Regresson model x1=0) LKGG Regresson model x1=1) Sy;x) Kaplan Meer x2=0) Kaplan Meer x2=1) LKGG Regresson model x2=0) LKGG Regresson model x2=1) Sy;x) Kaplan Meer x3=0) Kaplan Meer x3=1) LKGG Regresson model x3=0) LKGG Regresson model x3=1) y Fgure 5: Kaplan-Meer curves srafed by explanaory varable and esmaed survval funcons o he chemcal dependency daa: a) The maral saus explanaory varable. b) Schoolng explanaory varable. c) Age explanaory varable we perform he mehods suggesed by Cowles and Carln 1996). We use he beween and whn sequence nformaon, followng he approach developed n Gelman and Rubn 1992), o oban he poenal scale reducon, R. In all cases, hese values were close o one, ndcang he convergence of he chan. The approxmae poseror margnal densy funcons for he parameers are presened n Fgure 6. In Table 4, we repor poseror summares for he parameers of he LKGG model. We noe ha he values for he means a poseror Table 4) are que close as expeced) o he MLEs gven n Table 1. SD denoes he sandard devaon from he poseror dsrbuons of he parameers and HPD represens he 95% hghes poseror densy HPD) nervals. Densy Densy Densy λ λ Densy Densy Densy y ϕ ϕ Densy Densy Densy y Densy Densy Densy λ q q Densy Densy Densy ϕ β β Densy Densy Densy β β Densy Densy Densy q β2 Densy Densy Densy β0 β β3 Fgure 6: Approxmae poseror margnal denses for he parameers from β β he LKGG model for he chemcal dependency daa β2 β3

28 808 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. Table 4: Poseror summares for he parameers from he LKGG model for he chemcal dependency daa Parameer Mean SD HPD 95%) R q ; ) λ ; ) ϕ ; ) ; ) β ; ) β ; ) β ; ) β ; ) Concludng Remarks We nroduce an exended form of he Kumaraswamy generalzed gamma KGG) dsrbuon Pascoa e al., 2011) for whch he hazard rae funcon accommodaes he four ypes of shape forms,.e. ncreasng, decreasng, bahub and unmodal. The KGG model ncludes as specal cases several useful lfeme models. We derve explc expressons for s momens, generang funcon, order sascs and her momens. Furher, we also nroduce he called log- Kumaraswamy generalzed gamma LKGG) dsrbuon and oban explc expressons for s momens. Based on hs new dsrbuon, we defne he LKGG regresson model whch s very suable for modelng censored and uncensored lfeme daa. The new regresson model allow us o es as specal models he goodness of f of some wdely known regresson models. Hence, represens a good alernave for lfeme daa analyss. We esmae he model parameers usng maxmum lkelhood and a Bayesan approach. We demonsrae by means of an applcaon o real daa ha he LKGG model can produce beer fs han some well-known models. Acknowledgemens The auhors are graeful o he Assocae Edor and an anonymous referee and he Edor for very useful commens and suggesons. Ths work was suppored by CNPq and CAPES. Appendx A We derve an expanson for γ 1 k, x) λ 1 for any real λ > 0. We can wre

29 The Log-Kumaraswamy Generalzed Gamma Regresson Model 809 ) λ 1 γ 1 k, x) λ 1 = γ 1 k, x)}] λ 1 = 1) j γ 1 k, x)} j, j j=0 whch always converges snce 0 < γ 1 k, x) < 1. Hence, γ 1 k, x) λ 1 = j=0 m=0 j ) ) λ 1 j 1) j+m γ 1 k, x) m. 26) j m We can subsue j j=0 m=0 for m=0 j=m o oban where γ 1 k, x) λ 1 = s m λ 1) γ 1 k, x) m, 27) s m λ 1) = m=0 ) ) λ 1 j 1) j+m. 28) j m j=m We use he power seres for he ncomplee gamma rao funcon gven by γ 1 k, x) = xk Γk) d=0 x) d k + d) d!. By applcaon a resul by Gradsheyn and Ryzhk 2000, Secon 0.314) for a power seres rased o a posve neger q, we oban ) q a d x d = c q,d x d. 29) d=0 Here, he coeffcens c q,d for d = 1, 2, ) are deermned from he recurrence equaon c q,d = da 0 ) 1 d=0 d p q + 1) d] a p c q,d p, 30) p=1 where c q,0 = a q 0 and a p = 1) p /k + p)p!]. So, he coeffcen c q,d can be calculaed from c q,0,, c q,d 1, and hen can be expressed as a funcon of he quanes a 0,, a d, alhough s no necessary for programmng numercally he expansons. Furher, usng 29), we oban γ 1 k, x) q = xkq Γk) q c q,d x d. 31) d=0

30 810 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. Appendx B: Marx of second dervaves Lθ) Here, we provde formulas for he second-order paral dervaves of he loglkelhood funcon. Afer some algebrac manpulaons, we oban L, = υ 2 q 1 2 z 2 expqz )] + qz expqz )} q 1 q 2 1) q 1 Γq 2 ) w q 2 1 z expqz w )γ 1 q 2, w )] z expqz ) w 1 z }) + q 1 Γq 1 ) wq 2 z exp w )γ 1 q 2, w )] 1 λ 1 + λ ϕ)γ 1 q 2, w )] λ γ 1 q 2, w )] λ} 1 + w q 2 1 z )} expqz w )γ 1 q 2, w )] q 1 2 Γq 2 ) C w q 2 1 z expqz w ) γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} ϕ) q 1 q 2 1)z expqz ) q z + wq 2 Γq 2 ) exp w ) γ 1 q 2, w )] 1 λ λγ 1 q 2, w )] λ γ 1 q 2, w )] λ} 2 ϕ 1 + ϕ γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} )]}} ϕ) 1, L,λ = q 1 Γq 2 w q 2 1 z expqz w )γ 1 q 2, w )] 1 + ϕ) ) γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} λ logγ 1 q 2, w )] 1 + γ 1 q 2, w )] λ γ 1 q 2, w )] λ} )]} 1 ϕq 1 Γq 2 w q 2 1 z ) expqz w )γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} ϕ 1 γ 1 q 2, w )] λ} ϕ) λ logγ 1 q 2, w )] C

31 L,ϕ = The Log-Kumaraswamy Generalzed Gamma Regresson Model ϕ)γ 1 q 2, w )] λ γ 1 q 2, w )] λ} ) 1 λϕγ 1 q 2, w )] λ γ 1 q 2, w )] λ} 1 γ 1 q 2, w )] λ} } ϕ) 1, λq 1 Γq 2 ) w q 2 1 z expqz w )γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} 1 λq 1 Γq 2 w q 2 1 z expqz w ) ) C γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} ϕ 1 γ 1 q 2, w )] λ} ϕ) ϕ log γ 1 q 2, w )] λ} + γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} )} ϕ) 1, L,βj = q 2 x j 1 + qz ) expqz ) 1] q 1 2 Γq 2 ) γ 1 q 2, w )] 1 + λ 1 + ϕ)γ 1 q 2, w )] λ x j w q 2 1 z expqz ) γ 1 q 2, w )] λ} )] 1 q 1 q 2 1)w 1 z expqz ) qz ) exp w ) + q 2 q 2 1)w q 2 1 expqz ) + q 1 expqz 2w )γ 1 q 2, w )] 1 ] 1 Γq 2 ) wq 2 z q 1 Γq 2 ) wq 2 1 z expqz w ) γ 1 q 2, w )] λ} λγ 1 q 2, w )] 2λ 1 γ 1 q 2, w )] λ} )} 1 λ2 ϕ q 2 ) 2 Γq 2 )] 2 x j w q 2 1 expqz w ) C γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} ϕ 2 γ 1 q 2, w )] λ} ϕ) ϕ 1 + γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} )} ϕ) 1,

32 812 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. L λ,λ = υ λ 2 + ϕ) logγ1 q 2, w )] } 2 γ1 q 2, w )] λ γ 1 q 2, w )] λ} γ 1 q 2, w )] λ γ 1 q 2, w )] λ} } 1 + ϕ logγ1 q 2, w )] } 2 C γ 1 q 2, w )] λ γ 1 q 2, w )] λ} ϕ 1 γ 1 q 2, w )] λ} ϕ) γ 1 q 2, w )] λ γ 1 q 2, w )] λ} 1 ϕ 1 + γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} )]} ϕ) 1, L λ,ϕ = logγ 1 q 2, w )]γ 1 q 2, w )] λ γ 1 q 2, w )] λ} 1 + logγ 1 q 2, w )]γ 1 q 2, w )] 1 λ + γ 1 q 2, w )] λ} ϕ 1 C γ 1 q 2, w )] λ} ϕ) 1 ϕ log γ 1 q 2, w )] } 1 + γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} )]} ϕ) 1, L λ,βj = q 1 Γq 2 x j w q 2 1 expqz w )γ 1 q 2, w )] 1 + ϕ) ) γ 1 q 2, w )] λ γ 1 q 2, w )] λ} )} logγ 1 q 2, w )] q 1 Γq 2 x j w q 2 1 expqz w )γ 1 q 2, w )] λ 1 ) C γ 1 q 2, w )] λ} ϕ 1 γ 1 q 2, w )] λ} ϕ) logγ 1 q 2, w )] 1 + λγ 1 q 2, w )] λ γ 1 q 2, w )] λ} 1 ϕ 1 + γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} })]} ϕ) 1, L ϕ,ϕ = υ ϕ 2 C log γ 1 q 2, w )] λ}) 2 γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} ϕ) γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} } ϕ) 1,

33 L ϕ,βj = The Log-Kumaraswamy Generalzed Gamma Regresson Model 813 λq 1 Γq 2 ) x j w q 2 1 expqz w )γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} 1 λq 1 Γq 2 x j w q 2 1 expqz w ) ) γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} ϕ 1 γ 1 q 2, w )] λ} ϕ) log γ 1 q 2, w )] λ} ϕ 1 + γ 1 q 2, w )] λ} ϕ C γ 1 q 2, w )] λ} )]} ϕ) 1, L βj,β s = 1 2 x j x s expqz ) q 1 2 Γq 2 ) γ 1 q 2, w )] q + q 1 expqz ) x j x s w q 2 1 expqz w ) q 2 1)w 1 wq Γq 2 ) }) exp w )γ 1 q 2, w )] λ 1 + ϕ)γ 1 q 2, w )] λ γ 1 q 2, w )] λ} }) 1 λq 1 Γq 2 ) wq 2 expqz w ) γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} γ 1 q 2, w )] λ γ 1 q 2, w )] λ} )} 1 λϕq 1 2 Γq 2 ) C x j x s w q 2 1 expqz w ) γ 1 q 2, w )] λ 1 γ 1 q 2, w )] λ} ϕ 1 γ 1 q 2, w )] λ} ϕ) 1 q + q 1 expqz ) q 2 1)w 1 wq Γq 2 ) exp w ) γ 1 q 2, w )] λ 1 + γ 1 q 2, w )] λ γ 1 q 2, w )] λ} 1 ϕ 1 + γ 1 q 2, w )] λ} ϕ γ 1 q 2, w )] λ} )]})]} ϕ) 1, where z = y x T β)/ and w = q 2 expqz ).

34 814 Pascoa, M. A. R., de Pava, C. M. M., Cordero, G. M. and Orega, E. M. M. References Aars, R. M. 2000). Laurcella funcons. From MahWorld, A Wolfram Web Resource, creaed by Erc W. Wessen. hp://mahworld.wolfram.com/ LaurcellaFuncons.hml. Al, M. M., Woo, J. and Nadarajah, S. 2008). Generalzed gamma varables wh drough applcaon. Journal of he Korean Sascal Socey 37, Almpands, G. and Koropoulos, C. 2008). Phonemc segmenaon usng he generalzed gamma dsrbuon and small sample Bayesan nformaon creron. Speech Communcaon 50, Barkauskas, D. A., Kronewer, S. R., Lebrlla, C. B. and Rocke, D. M. 2009). Analyss of MALDI FT-ICR mass specromery daa: a me seres approach. Analyca Chmca Aca 648, Baker, T. B., Pper, M. E., McCarhy, D. E., Majeske, M. R. and Fore, M. C. 2004). Addcon movaon reformulaed: an effecve processng model of negave renforcemen. Psychologcal Bullen 111, Brandon, T. H., Vdrne, J. I. and Lvn, E. B. 2007). Relapse and relapse prevenon. Annual Revew of Psychology 3, Cancho, V. G., Bolfarne, H. and Achcar, J. A. 1999). A Bayesan analyss for he exponenaed Webull dsrbuon. Journal of Appled Sascs 8, Cancho, V. G., Orega, E. M. M. and Bolfarne, H. 2009). The exponenaed- Webull regresson models wh cure rae. Journal of Appled Probably & Sascs 4, Cowles, M. K. and Carln, B. P. 1996). Markov chan Mone Carlo convergence dagnoscs: a comparave revew. Journal of he Amercan Sascal Assocaon 91, Carrasco, J. M. F., Orega, E. M. M. and Cordero, G. M. 2008). A generalzed modfed Webull dsrbuon for lfeme modelng. Compuaonal Sascs and Daa Analyss 53, Cordero, G. M., Orega, E. M. M. and Slva, G. O. 2011). The exponenaed generalzed gamma dsrbuon wh applcaon o lfeme daa. Journal of Sascal Compuaon and Smulaon 81,

35 The Log-Kumaraswamy Generalzed Gamma Regresson Model 815 Cox, D. R. 1972) Regresson models and lfe-ables wh dscusson). Journal of he Royal Sascal Socey, Seres B 34, Cox, C. 2008). The generalzed F dsrbuon: an umbrella for paramerc survval analyss. Sascs n Medcne 27, Cox, C., Chu, H., Schneder, M. F. and Muñoz, A. 2007). Paramerc survval analyss and axonomy of hazard funcons for he generalzed gamma dsrbuon. Sascs n Medcne 26, Exon, H. 1978). Handbook of Hypergeomerc Inegrals: Theory, Applcaons, Tables, Compuer Programs. Halsed Press, New York. Gelman, A. and Rubn, D. B. 1992). Inference from erave smulaon usng mulple sequences wh dscusson). Sascal Scence 7, Gomes, O., Combes, C. and Dussauchoy, A. 2008). Parameer esmaon of he generalzed gamma dsrbuon. Mahemacs and Compuers n Smulaon 79, Gradsheyn, I. S. and Ryzhk, I. M. 2000). Table of Inegrals, Seres, and Producs, 6h edon. Eded by A. Jeffrey and D. Zwllnger. Academc Press, New York. Gupa, R. D. and Kundu, D. 1999). Generalzed exponenal dsrbuons. Ausralan & New Zealand Journal of Sascs 41, Hashmoo, E. M., Orega, E. M. M., Cancho, V. G. and Cordero, G. M. 2010). The log-exponenaed Webull regresson model for nerval-censored daa. Compuaonal Sascs & Daa Analyss 54, Kalvas, P. W. and Volkow, N. D. 2005). The neural bass of addcon: a pahology of movaon and choce. Amercan Journal Psycharc 162, Koob, G. F. and Le Moal, M. 1997). Drug abuse: hedonc homeosac dysregulaon. Scence 278, Kundu, D. and Raqab, M. Z. 2005). Generalzed Raylegh dsrbuon: dfferen mehods of esmaon. Compuaonal Sascs and Daa Analyss 49, La, C. D., Xe, M. and Murhy, D. N. P. 2003). A modfed Webull dsrbuon. IEEE Transacons on Relably 52,