Bayesian Estimation of the Kumaraswamy Inverse Weibull Distribution

Transcription

1 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) Bayesan Esmaon of he Kumaraswamy Inverse Webull Dsrbuon Felpe R S de Gusmão Deparmen of Sascs, Federal Unversy of São Carlos, Brazl felpe556@gmalcom Vera L D Tomazella Deparmen of Sascs, Federal Unversy of São Carlos, Brazl vera@ufscarbr Rcardo S Ehlers Deparmen of Appled Mahemacs and Sascs, Unversy of São Paulo, Brazl ehlers@cmcuspbr Receved 25 Sepember 215 Acceped 23 Augus 216 The Kumaraswamy Inverse Webull dsrbuon has he ably o model falure raes ha have unmodal shapes and are que common n relably and bologcal sudes The hree-parameer Kumaraswamy Inverse Webull dsrbuon wh decreasng and unmodal falure rae s nroduced We provde a comprehensve reamen of he mahemacal properes of he Kumaraswany Inverse Webull dsrbuon and derve expressons for s momen generang funcon and he r-h generalzed momen Some properes of he model wh some graphs of densy and hazard funcon are dscussed We also dscuss a Bayesan approach for hs dsrbuon and an applcaon was made for a real daa se Keywords: Kumaraswamy dsrbuon; Webull dsrbuon; Survval; Bayesan analyss 2 Mahemacs Subjec Classfcaon: 22E46, 53C35, 57S2 1 Inroducon In hs paper we propose a new probably dsrbuon o handle he problem of survval daa Movaed by research developed n recen years, we nroduce he Kumaraswany Inverse Webull dsrbuon ha ncludes several well known dsrbuons used n survval analyss Recenly, many auhors have proposed new classes of dsrbuons, whch are modfcaons of dsrbuon funcons whch provde hazard raos conemplang varous shapes We can ce for example he Webull exponenal 6], whch has also he hazard rae funcon wh a unmodal form, see also 13]) 1] proposed a four-parameer dsrbuon denoed generalzed modfed Webull Copyrgh 217, he Auhors Publshed by Alans Press Ths s an open access arcle under he CC BY-NC lcense hp://creavecommonsorg/lcenses/by-nc/4/) 248

2 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) GMW) dsrbuon, 3] nroduced and suded he r-paramerc nverse Webull generalzed dsrbuon ha possesses falure rae wh unmodal, ncreasng and decreasng forms 9] proposed a dsrbuon wh four parameers, called bea generalzed half normal dsrbuon Underexplored n he leraure and rarely used by sascans, he Kumaraswamy dsrbuon 8] has a doman n he real nerval,1) Ths propery urns he Kumaraswamy dsrbuon a naural canddae o combne wh oher dsrbuons o produce a more general one Is cumulave dsrbuon funcon cdf) s gven by, and s probably densy funcon pdf) s gven by, F x;a,b) = 1 1 x a ] b, 11) f x;a,b) = abx a 1 1 x a ] b 1, < x < 1, where a > and b > Ths densy can be unmodal, ncreasng, decreasng or consan Recenly, 2] proposed o use he Kumaraswamy o generalze oher dsrbuons Consderng ha a random varable X has dsrbuon G, hey sugges o apply he Kumarasawamy dsrbuon o Gx) Noe ha, snce < Gx) < 1 for any dsrbuon funcon G, hen evaluang Eq 11) a Gx) we oban, F G x;a,b) = 1 1 Gx) a ] b 12) where F G s he cdf of he generalzed Kumaraswamy-G dsrbuon Based on hese deas, we consder he Inverse Webull Dsrbuon as a canddae for G, usng Eq 12) Then, performng some adjusmens and mahemacal manpulaons, we oban he Kumaraswamy nverse Webull Kum-IW) dsrbuon The res of he paper s organzed as follows In Secon 2, we develop he Kum-IW dsrbuon Secon 3 s devoed o descrbe basc properes of he dsrbuon Inference procedures va maxmum lkelhood and Bayesan approaches are presened n Secon 4 Secon 5 s devoed o analyze a real daa se and n Secon 6 we presen some conclusons of hs work 2 Kumaraswamy nverse Webull dsrbuon Le T a random varable wh nverse Webull dsrbuon Then s cdf can be wren as, α ) ] G) = exp, >, 21) where α >, >, and s pdf s gven by, g) = α +1) exp α ) ] Inserng he G funcon of Eq 21) n Eq 12) follows ha, { α ) ]} b F G ;a,b,α,) = 1 1 exp a 22) We noe ha he parameers a and α n 22) are no denfable and we adop he reparameerzaon c = αa 1/ so ha he Kum-IW cdf s rewren as, { c ) ]} b F G ;b,c,) = 1 1 exp, 23) 249

3 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) where b > and c > are he shape and scale paremeers respecvely Accordngly, he Kum-IW pdf s now gven by, c ) ]{ c ) ]} b 1 f G ;b,c,) = bc +1) exp 1 exp 24) I can be easly seen ha when b = 1 we oban he pdf of he Inverse Webull IW) dsrbuon gven by, α ) ] g) = α +1) exp Fnally, he correspondng survval and hazard funcons are respecvely gven by, { c ) ]} b S G ;b,c,) = 1 exp and h G ;b,c,) = whle he quanle funcon, Qu), of he Kum-IW dsrbuon s gven by, bc +1) exp 1 exp c ) ] c ) ] Qu) = F 1 u;b,c,) = c log1 1 u) 1 b )) 1 21 Some specal classes of he Kum-IW The followng well known and new dsrbuons are specal sub-classes of he Kum-IW dsrbuon Kumaraswamy nverse Raylegh dsrbuon Kum-IR) If = 2, he Kum-IW dsrbuon reduces o he Kumaraswamy nverse Raylegh dsrbuon Kum-IR) Then, wh = 2 he densy funcon of Kum-IW s expressed by: { c ) ]} 2 b F G ;b,c) = 1 1 exp, where b > s he shape parameer, and c > s he scale parameer Hence, he KR dsrbuon has wo parameers, and s pdf s gven by c ) ]{ 2 c ) ]} 2 b 1 f G ;b,c) = 2bc 2 3 exp 1 exp The correspondng survval and hazard funcons are gven respecvely by, { c ) ]} 2 b S G ;b,c) = 1 exp and h G ;b,c) = Inverse Raylegh dsrbuon IR) 2bc 2 3 exp 1 exp ) ] c 2 ) ] 2 c 25

4 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) If = 2 and b = 1, he Kum-IW dsrbuon reduces o he nverse Raylegh dsrbuon Kum-IR) Then, wh = 2 and b = 1 he densy funcon of Kum-IW s expressed by: α ) ] 2 G) = exp, >, where α >, and s pdf s α ) ] 2 g) = 2α 2 3 exp Kumaraswamy nverse Exponenal dsrbuon Kum-IE) If = 1, he Kum-IW dsrbuon reduces o he Kumaraswamy nverse Exponenal dsrbuon Kum-IE) Then, wh = 1 he densy funcon of Kum-IW s expressed by: { c )]} b F G ;b,c) = 1 1 exp, where b > s he shape parameer, and c > s he scale parameer Hence, he KIE dsrbuon has wo parameers, and s pdf s gven by c )]{ c )]} b 1 f G ;b,c) = bc 2 exp 1 exp The correspondng survval and hazard funcons are respecvely { c )]} b S G ;b,c) = 1 exp and h G ;b,c) = bc 2 exp )] c 1 exp )] c Inverse Exponenal dsrbuon IE) If = 1 and b = 1, he Kum-IW dsrbuon reduces o he Inverse Exponenal dsrbuon IE) Then, wh = 1 and b = 1 he densy funcon of Kum-IW s expressed by: )] λ G) = exp, >, where λ > and s pdf s )] λ g) = λ 2 exp 3 Basc Properes In hs secon we descrbe n deal some properes lke expansons, momens, mean devaons, Bonferron and Lorenz curves, order sascs and enropes whch mgh be useful n any applcaon of he dsrbuon 251

5 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) f) b = 2 ; c = 1 ; = 1 b = 3 ; c = 26 ; = 28 b = 4 ; c = 48 ; = 23 b = 5 ; c = 62 ; = 25 f) b = 22 ; c = 8 ; = 1 b = 35 ; c = 23 ; = 28 b = 5 ; c = 4 ; = 23 b = 4 ; c = 72 ; = a) b) Fg 1 Kumaraswamy nverse Webull densy funcons S) b = 2 ; c = 1 ; = 15 b = 3 ; c = 29 ; = 18 b = 4 ; c = 49 ; = 22 b = 5 ; c = 62 ; = 25 S) b = 22 ; c = 1 ; = 15 b = 2 ; c = 4 ; = 18 b = 4 ; c = 47 ; = 25 b = 55 ; c = 7 ; = a) b) Fg 2 Kumaraswamy nverse Webull survval funcons 31 Expansons for he dsrbuon and densy funcons We now gve smple expansons for he cdf of he Kumaraswamy Inverse Webull dsrbuon If x < 1 and ψ > s a non-neger real number, we have 1 x) ψ = = 1) ψ!)x 31) 252

6 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) h) b = 2 ; c = 1 ; = 25 b = 3 ; c = 25 ; = 3 b = 4 ; c = 82 ; = 35 b = 12 ; c = 12 ; = 1 h) b = 2 ; c = 5 ; = 1 b = 35 ; c = 23 ; = 28 b = 5 ; c = 4 ; = 23 b = 4 ; c = 72 ; = a) b) Fg 3 Kumaraswamy nverse Webull hazard funcons If ψ s a posve neger, he seres sops a = ψ Usng expanson n Eq 31) follows ha, f ;b,c,) = bc +1) 1) Γb) c ) ]!)Γb ) exp + 1) 32) and F ;b,c,) = 1 = = 1) Γb + 1)!)Γb + 1 ) exp c ) ] Because he negrals nvolved n he compuaon of momens, Bonferron and Lorenz curves, relably, Shannon and Rény enropes and oher nferenal resuls do no have analycal soluons, hese expansons are necessary 32 A general formula for he momens of he Kum-IW We hardly need o emphasze he need and mporance of he momens n any sascal analyses, especally n appled work Some of he mos mporan feaures and characerscs of a dsrbuon can be suded usng her momens eg endency, dsperson, skewness and kuross) If he random varable T follows he Kum-IW dsrbuon, s k-h momen abou zero s gven by, E k) = k bc +1) exp = bc k c ) ] 1) r Γb) Γb r)r! r + 1) k 1 Γ The momen generang funcon Mz) of T for z < 1 s, M z) = { n k= z k k! bck 1 exp ) 1 k c ) ]] b 1 d 1) r Γb) Γb r)r! r + 1) k 1 Γ 1 k ) } 253

7 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) Hence, for z < 1, he cumulave generang funcon of T s K z) = log n k= k! bck 1)r Γb) Γb r)r! r + 1) k 1 Γ 1 k )] ] z k We noe ha was necessary o use he expansons prevously presened for he resuls of hs secon 33 Mean devaons The amoun of scaerng n a populaon may be measured by all he absolue values of he devaons from he mean or he medan If X s a random varable wh Kum-IW dsrbuon wh mean µ = EX] and medan M, hen he average devaon from he average and he average devaon o he medan are defned respecvely by, δ 1 X) = x µ f x)dx and δ 2 X) = Usng he densy of exended Kum-IW and gven ha, µ x f x)dx = bc 1) r ) 1 Γb) 1 1 γ Γb r)r! 1 + r x M f x)dx 1 where γa,x) s he lower ncomplee Gamma funcon, follows ha 1) δ 1 X) = 2µFµ) 2µ + 2bc r ) 1 Γb) 1 1 γ Γb r)r! 1 + r ), c 1 + r), µ 1 ), c 1 + r) µ and δ 2 X) = 2µFµ) 2µ + 2bc 1) r ) 1 Γb) 1 1 γ Γb r)r! 1 + r 1 ), c 1 + r) M 34 Bonferron and Lorenz curves Bonferron and Lorenz curves are wdely appled no only n economcs o sudy ncome and povery, bu also n oher felds such as relably, demography, nsurance and medcne Le hen µ = EX) e q = F 1 p;θ), where F 1 ) s he nverse funcon of he cumulave funcon of a random varable X Bonferron and Lorenz curves are defned by, Then, usng he expanded densy, q Bp) = 1 q x f x)dx and Lp) = 1 q x f x)dx pµ µ x f x)dx = bc 1) r ) 1 Γb) 1 1 Γ Γb r)r! 1 + r 1 where Γa,x) s he upper ncomplee gamma funcon Therefore, we have, ), c 1 + r), q 254

8 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) he Bonferron curve whch s gven by: Bp) = bc pµ he Lorenz curve whch s gven by: Lp) = bc µ 1) r ) 1 Γb) 1 1 Γ Γb r)r! 1 + r 1) r ) 1 Γb) 1 1 Γ Γb r)r! 1 + r 1 1 ), c 1 + r) ; q ), c 1 + r) q 35 Order sascs and Shannon enropy Le T 1:n T 2:n T n:n be he order sascs obaned from he Kum-IW dsrbuon The random varable T r:n, for r = 1,,n, denoes he r-h order sasc n a sample of sze n The pdf of T r:n wren as, f r:n ) = C r:n F) r 1 1 F)] n r f ), >, where f ) and F) are gven by Eq 24) and 23) and C r:n = n!/r 1)!n r)!] The k-h momen µ k) r:n of he rh order sasc s for k = 1,2,, and 1 r n Hence, µ r:n k) = ETr:n) k = k C r:n F)] r 1 1 F)] n r f )d, µ k) r:n = C r:n 1 and we oban an expresson for he momen gven by, ET k r:n) = C r:n bc j= = j n r 1 j) n 1 d j 1) + j Γbn br + b j + b)! j!γr j)γbn br + b ) + 1) k 1 Γ 1 k ) The Shannon enropy of a random varable T s defned as a measure of he quany of nformaon A ceran message has more quany of nformaon he greaer degree of uncerany and s defned mahemacally by E{ log f )]}, where f ) s he fdp of T In parcular, for a random varable T whch follows he Kum-IW dsrbuon we have, ) E { log f )]} = log bc + b + 1)logc) 1 b + 1)b 1) r Γb) Γb r)r! = 1) r Γb) Γb r)r! 1 γ logr + 1)] + c 1 r + 1) 1 r + 1) 1) r Γb) Γb r)r! r + 1) 1 1 Γ 1 1 ) + 1) +r Γb)Γb + 1)!r!Γb r)γb + 1 j) j + 1)2, j > 1 33) where γ = log j)exp{ j}d j s he approxmae value of he Euler s consan 255

9 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) Rény enropy The enropy of a random varable X wh densy funcon 32) measurng he uncerany of he varaon The Rény enropy s gven by, I r ρ) = 1 { 1 ρ log f x) dx} ρ where ρ > and ρ 1 In nformaon heory, Rény enropy generalzes he Shannon enropy Ths form of enropy s mporan especally n ecology and sascs, where can be used as an ndex of dversy In quanum nformaon, can be used as a measure of enanglemen If X s a random varable and follows he Kum-IW dsrbuon, hen he Rény enropy s gven by I r ρ) = 1 1 ρ log 4 Inference for Censored Daa { ρ 1 bc 1 ρ 1) Γρb ρ + 1)!)Γρb ρ + 1 ) r + ) 1 ρ = 41 Maxmum lkelhood esmaon ) 1+ 1 ρ Γ + ρ 1 ) } Le T be a random varable wh Kum-IW dsrbuon wh parameer vecor θ = b,c,) The daa n survval analyss and relably sudes are generally censored A very smple random censorng mechansm ha s ofen realsc s one n whch each ndvdual s assumed o have a lfeme T and a censorng me C, where T and C are ndependen random varables Suppose he daa se consss of n ndependen observaons = mnt,c ) for = 1,,n The dsrbuon of C does no depend on any of he unknown parameers of T Paramerc nference for such daa are usually based on lkelhood mehods and her asympoc heory The censored log-lkelhood lθ) for he model parameers s ) ) 1 ) ]] c lθ) = r log bc c + 1) F log ) + b 1) log 1 exp + F F b log C 1 exp ) ]] c, 41) where F = 1,r] and C = r + 1,n]; sll, C represens he censored daa and F represens he falure daa The maxmum lkelhood esmae MLE) θ of θ s obaned by solvng he nonlnear lkelhood equaons U b θ) = lθ) b =, U c θ) = lθ) c = and U θ) = lθ) = These equaons canno be solved analycally and sascal sofware can be used o solve he equaons numercally For nerval esmaon of b, c and, and ess of hypoheses on hese parameers, we mus oban he 3 3 observed nformaon marx Jθ) whch s gven by, J bb θ) J bc θ) J b θ) Jθ) = J cb θ) J cc θ) J c θ), J b θ) J c θ) J θ) 256

10 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) Under condons me for parameers obeyng he paramerc space and no consderng he lms of he same, he asympoc dsrbuon of n θ θ) s N 3,Iθ) 1 ), where Iθ) s he expeced nformaon marx Ths asympoc behavor s vald f Iθ) s replaced by J θ), he observed nformaon marx evaluaed a θ The asympoc mulvarae normal dsrbuon N 3,J θ) 1 ) can be used o consruc approxmae confdence nervals and confdence regons for he ndvdual parameers and for he hazard and survval funcons The asympoc normaly s also useful for esng goodness of f of he hree parameers he Kum-IW dsrbuon and for comparng hs dsrbuon wh some of s specal submodels usng one of he wo well-known asympocally equvalen es sascs - namely, he lkelhood rao LR) sasc and he Wald and Rao score sascs 42 Bayesan approach Followng he Bayesan paradgm, we need o complee he model specfcaon by specfyng a pror dsrbuon for he parameers By Bayes Theorem, he poseror dsrbuon s hen proporonal o he produc of he lkelhood funcon by he pror densy Subjevsm s he predomnan phlosophcal foundaon n Bayesan nference, alhough n pracce nonnformave pror denses bul on some formal rule) are frequenly used 7]) Snce he parameers n he Kum-IW dsrbuon are all posve quanes and due o he flexbly generaed by he wo-parameer Gamma dsrbuon hs s adoped as pror dsrbuon So, b Gammam,w), c Gammaa,s) and Gammax,l) Assumng ndependence among he pror denses, he poseror densy s expressed by, ] hθ ) b m+n 1 c a+n 1 x+n 1 exp n =1 1 wb sc l c n =1 { ) ]} c b 1 1 exp 42) Ths jon densy has no known analycal form bu we can provde an approxmae soluon based on he complee condonal dsrbuons of b, c and These are gven by he followng expressons, hb c,,) b m+n 1 exp hc b,,) c a+n 1 exp h b,c,) x+n 1 exp wb c sc c l c n =1 n =1 n =1 ] ] ] n =1 n =1 n = { c 1 exp { c 1 exp { c 1 exp ) ]} b 1, ) ]} b 1, ) ]} b 1 257

11 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) Applcaon In hs secon, we presen esmaon resuls for parameers of he Kum-IW dsrbuon under a Bayesan approach usng a real daa se The commercal producon of cale mea n Brazl, whch usually comes from cale of he Nelore race, seeks o opmze he process ryng o oban a me for he cale o reach he specfc wegh n he perod from he brh unl weans For a daa se wh 69 bulls of he Nelore race, he mes n days) unl he anmals reached he wegh of 16kg relave o he perod from brh unl weans were observed We hen compared he classcal Kaplan-Meyer and Bayesan survval funcons hrough wo graphc mehods We used Markov chan Mone Carlo MCMC) smulaon o esmae he parameers θ = b,c,) T of he Kum-IW dsrbuon Usng he expresson for he log-lkelhood lθ) and Gamma prors for he parameers a roune was wren for he package Wnbugs see 11]) The resuls appear n Table 1 n erms of poseror mean, sandard devaon SD), medan and 95% credble nervals of he hree parameer Fgure 4 show he race plos of smulaed parameer values and esmaes of margnal poseror denses Table 1 Resuls of he Bayesan approach o he Kum-IW dsrbuon Parameer Mean SD 25% Medan 975% b c One of he mehods a our dsposal o check f our model s well adjused o he daa consss of a comparson of he survval funcon of he proposed paramerc model wh he Kaplan-Meer esmaor Anoher mehod consss of skechng he survval funcon of he paramerc model versus he Kaplan-Meer esmae for he survval funcon, f hs curves s close o he sragh lne y = x we wll have a good adjusmen see Fgure 5) 6 Conclusons We worked ou a hree parameer lfeme dsrbuon called he Kumaraswamy Inverse Webull Kum-IW) dsrbuon whch exends Inverse Webull dsrbuon proposed and wdely used n he lfeme leraure The model s much more flexble han he nverse Webull The Kum-IW dsrbuon could have ncreasng, decreasng and unmodal hazard raes We provde a mahemacal overvew of hs dsrbuon ncludng he denses of he order sascs, Rény enropy, Shannon enropy, Bonferron and Lorenz curves and Mean devaons Also, we derve an explc algebrac formula for he r-h momen, expressons for he order sascs, and he maxmum lkelhood esmaon for he censored daa The performance of he model was analzed usng real daa ses where he Kum-IW dsrbuon performng very well and he esmaon was gven by Bayes mehod 258

12 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) Trace of c Trace of bea Trace of b Ieraons Densy of b Densy of c Densy of bea 2 N = Ieraons 2 Ieraons 5 14 Bandwdh = N = Bandwdh = N = Bandwdh = Fg 4 Trace plos of smulaed parameer values of he Kumaraswamy nverse Webull dsrbuon Lnear Bayesan 2 4 S) S) 6 8 Kaplan Meer Bayesan me weeks) S) Kaplan Meer a) b) Fg 5 a) Comparson beween survval funcons generaed by he Bayesan mehod and Kaplan-Meer esmaor descrbed by plong S) versus me b) Comparson beween survval funcons generaed by he Bayesan mehod descrbed by plong S) versus Kaplan-Meer esmae 259

13 Journal of Sascal Theory and Applcaons, Vol 16, No 2 June 217) References 1] Carrasco, J F, Orega, E M M, and Cordero, G M 28) A generalzed modfed Webull dsrbuon for lfeme modelng Compuaonal Sascs & Daa Analyss, 532): ] Cordero, G M and de Casro, M 211) A new famly of generalzed dsrbuons Journal of Sascal Compuaon and Smulaon, 817): ] Gusmão, F R S, Orega, E M M, and Cordero, G M 211) The generalzed nverse Webull dsrbuon Sascal Papers, 52: ] Gusmão, F R S, Orega, E M M, and Cordero, G M 212) Reply o he Leer o he Edor of M C Jones Sascal Papers, 53: ] Gupa, RD, Kundu, D 1999) Generalzed exponenal dsrbuon Ausralan and New Zealand Journal of Sascs 412): ] Mudholkar, G S, Srvasava, D K, and Fremer, M 1995) The exponenaed Webull famly: A reanalyss of he bus-moor-falure daa Technomercs, 374): ] Kass, RE, Wasserman, L 1996) The Selecon of Pror Dsrbuons by Formal Rules Journal of he Amercan Sascal Assocaon, 91: ] Kumaraswamy, P 198) A generalzed probably densy funcon for double-bounded random processes Journal of Hydrology, 461?2): ] Pescm, R R, Deméro, C G B, Cordero, G M, Orega, E M M, and Urbano, M R 21) The bea generalzed half-normal dsrbuon Compuaonal Sascs & Daa Analyss, 544): ] Pollard, W E 1986) Bayesan sascs for evaluaon research An Inroducon Sage Publcaons New Delh, 241p 11] Spegelhaler, D, Thomas, A, Bes, N, Lunn, D 27) Wnbugs user manual, Verson ] R Core Team 212) R: A Language and Envronmen for Sascal Compung R Foundaon for Sascal Compung, Venna, Ausra ISBN: ] Xe, M and La, C D 1996) Relably analyss usng an addve Webull model wh bahub-shaped falure rae funcon Relably Engneerng & Sysem Safey, 521):