The generalized weighted Lindley distribution: Properties, estimation, and applications

Transcription

1 Ramos & Louzada, Coget Mathematcs 6, : 56 STATISTICS RESEARCH ARTICLE The geeralzed weghted Ldley dstrbuto: Propertes, estmato, ad applcatos PL Ramos * ad F Louzada Receved: July 6 Accepted: 5 October 6 Frst Publshed: November 6 *Correspodg author: PL Ramos, Isttute of Mathematcal Scece ad Computg, Uversdade de São Paulo, Sao Carlos-SP, Brazl E-mal: pedrolramos@uspbr Revewg edtor: Hrosh Shrash, Keo uversty, Japa Addtoal formato s avalable at the ed of the artcle Abstract: I ths study, a three-parameter lfetme dstrbuto amely geeralzed weghted Ldley GLW dstrbuto s proposed The GLW dstrbuto s a useful geeralzato of the weghted Ldley dstrbuto whch accommodates creasg, decreasg, decreasg-creasg-decreasg, bathtub, ad umodal hazard rate makg t a flexble model for relable data A sgfcat accout of mathematcal propertes for ths dstrbuto s preseted Dfferet estmato procedures are dscussed such as maxmum lkelhood estmators, method of momets, ordary ad weghted least-squares, percetle, maxmum product of spacgs, ad mmum dstace estmators The estmators are compared by extesve umercal smulatos Fally, two data-sets are aalyzed for llustratve purposes provg that the GWL outperforms several other three-parameter lfetme dstrbutos Subjects: Scece; Mathematcs & Statstcs; Statstcs & Probablty; Statstcs; Statstcal Computg; Statstcal Theory & Methods Keywords: geeralzed weghted Ldley dstrbuto; deret estmato methods; maxmum product of spacgs Itroducto I recet years, several ew extesos of the expoetal dstrbuto have bee troduced the lterature for descrbg real problems Ghtay, Ateh, ad Nadarajah 8 vestgated dfferet propertes of the Ldley dstrbuto ad outled that may cases the Ldley dstrbuto ABOUT THE AUTHORS P L Ramos holds a BSc degree Statstcs ad a MSc Appled ad Computatoal Mathematcs from the São Paulo State Uversty, Brazl He s curretly readg for hs PhD Statstcs at the Isttute for Mathematcal Scece ad Computg, Uversty of São Paulo USP, Brazl Hs ma research terests are survval aalyss, Bayesa ferece, classcal ferece, ad probablty dstrbuto theory F Louzada s a professor of Statstcs at the Isttute for Mathematcal Scece ad Computg, Uversty of So Paulo USP, Brazl He receved hs PhD degree Statstcs from the Uversty of Oxford, UK, hs MSc degree Computatoal Mathematcs from USP, Brazl, ad hs BSc degree Statstcs from UFSCar, Brazl Hs ma research terests are survval aalyss, data mg, Bayesa ferece, classcal ferece, ad probablty dstrbuto theory PUBLIC INTEREST STATEMENT We have proposed ad preseted a probablty dstrbuto called geeralzed weghted Ldley WL dstrbuto Ths dstrbuto s a useful geeralzato of the WL dstrbuto whch accommodates creasg, decreasg, decreasgcreasg-decreasg, bathtub, ad umodal hazard rate A sgfcat accout of mathematcal propertes for ths dstrbuto was preseted Dfferet estmato procedures were proposed ad compared by extesve umercal smulatos We beleve that ew dstrbuto wll allow the users to descrbe dfferet data-sets obtag a better predctve performace comparso wth other usual dstrbutos 6 The Authors Ths ope access artcle s dstrbuted uder a Creatve Commos Attrbuto CC-BY lcese Page of 8

2 Ramos & Louzada, Coget Mathematcs 6, : 56 outperforms expoetal dstrbuto Sce the, may geeralzatos of the Ldley dstrbuto have bee troduced such as geeralzed Ldley Zakerzadeh & Dolat, 9, exteded Ldley Bakouch, Al-Zahra, Al-Shomra, March, & Louzada,, expoetal Posso Ldley Barreto- Souza & Bakouch,, ad Power Ldley Ghtay, Al-Mutar, Balakrsha, & Al-Eez, dstrbuto Ghtay, Alqallaf, Al-Mutar, ad Husa troduced a ew class of weghted Ldley WL dstrbuto addg more flexblty to the Ldley dstrbuto Let T be a radom varable wth a WL dstrbuto The probablty desty fucto pdf s gve by f t λ, φ for all t >, φ > ad λ > ad Γφ e x x φ dx s the gamma fucto Oe of ts peculartes s that the hazard fucto ca have a creasg φ or bathtub <φ< shape Dfferet propertes ad estmato methods for ths model were preseted by Mazuchel, Louzada, ad Ghtay, Al 5, Wag ad Wag press, Al-Mutar, Ghtay, ad Kudu 5 I ths study, a ew lfetme dstrbuto famly s proposed whch s a drect geeralzato of the WL dstrbuto The pdf s gve by f t φ, λ, λ φ+ λ + φγφ tφ + te λt, λ φ λ + φγφ tφ λ +λt e λt, for all t >, φ >, λ> ad > Importat probablty dstrbutos ca be obtaed from the GWL dstrbuto as the WL dstrbuto, Power Ldley dstrbuto φ ad the Ldley dstrbuto φ ad Due to ths relatoshp, such model could also be amed as weghted power Ldley or geeralzed power Ldley dstrbuto Torab, Falahat-Nae, ad Motazer dscussed a class of dstrbuto wth four parameters whch s a geeralzato of the proposed model Such dstrbuto cludes the geeralzed WL, geeralzed gamma GG dstrbuto, gamma ad Webull, amog others The ma dfferece of ths study les the fact that the proposed three-parameter dstrbuto has a smple structure wth less computatoal ssues I ths way, the behavor of the pdf ad the hazard fucto ca be studed Ths model has dfferet forms of hazard fucto such as creasg, decreasg, bathtub, umodal, or decreasg-creasg-decreasg shape makg the GWL dstrbuto a flexble model for relable data Moreover, a sgfcat accout of mathematcal propertes for the ew dstrbuto s provded The feretal procedures for the parameters of GLW dstrbuto are preseted cosderg dfferet methods such as maxmum lkelhood estmators MLE, methods of momets ME, ordary least-squares estmato OLSE, weghted least-squares estmato WLSE, maxmum product of spacgs MPS, Cramer-vo Mses type mmum dstace CME, Aderso Darlg ADE ad rghttal Aderso Darlg RADE The performace of these estmato procedures are compared usg extesve umercal smulatos Fally, two data-sets are aalyzed for llustratve purposes provg that the GWL outperforms several usual three-parameter lfetme dstrbutos such as the GG dstrbuto Stacy, 96, the geeralzed Webull GW dstrbuto Mudholkar, Srvastava, & Kolla, 996, the geeralzed expoetal-posso GEP dstrbuto Barreto-Souza & Crbar-Neto, 9, ad the expoetated Webull EW dstrbuto Mudholkar, Srvastava, & Fremer, 995 The results of ths paper are orgazed as follows Secto provdes a sgfcat accout of mathematcal propertes for the ew dstrbuto Secto presets the eght estmato methods whch are cosdered I the Secto, a smulato study s preseted order to detfy the most effcet procedure Secto 5 llustrates the proposed methodology two real data-sets Secto 6 summarzes the preset work Page of 8

3 Ramos & Louzada, Coget Mathematcs 6, : 56 Fgure Desty fucto shapes for GWL dstrbuto cosderg dfferet values of φ, λ ad ft 5 5 φ, λ5, φ5, λ5, 5 φ, λ5, φ, λ5, 5 φ5, λ, ft 5 5 φ, λ5, φ5, λ5, 6 φ, λ5, 8 φ5, λ5, 5 φ, λ5, t t Geeralzed weghted Ldley dstrbuto The geeralzed WL dstrbuto ca be expressed as a two-compoet mxture f t φ, λ, pf t φ, λ, + pf t φ, λ, where p λ λ + φ ad T j GGφ + j, λ,, for j,, e f j t λ, φ has GG dstrbuto, gve by f j t φ, λ, Γφ + j λφ+j t φ+j e λt The behavor of the pdf whe t ad t are, respectvely, gve by, f φ < λ f, f φ λ + φγφ, f φ >, quadf Fgure gves examples of the shapes of the desty fucto for dfferet values of φ, λ ad The cumulatve dstrbuto fucto from the GWL dstrbuto s gve by Ft φ, λ, γ[ φ, λt ] λ + φ λt φ e λt λ + φγφ where γ[y, x] x wy e w dw s the lower complete gamma fucto Momets May mportat features ad propertes of a dstrbuto ca be obtaed through ts momets such as mea, varace, kurtoss, ad skewess I ths secto, mportat momet fuctos such as the momet-geeratg fucto, r-th momet, r-th cetral momet, amog others are preseted Theorem For the radom varable T wth GWL dstrbuto, the momet-geeratg fucto s gve by M X t r t r λ r r! r + φ + λ Γ r + φ λ + φγφ 5 Page of 8

4 Ramos & Louzada, Coget Mathematcs 6, : 56 Proof M X,j t Note that, the momet-geeratg fucto from GG dstrbuto s gve by t r Γ r + φ + j r! λ r Γφ + j r Sce the GWL dstrbuto ca be expressed as a two-compoet mxture, we have M X t E [ e tx] λ λ + φ λ + φ r t r λ r r! e tx f x φ, λ, dx pm X, t+ pm X, t r r t r Γ r + φ + r! λ r Γφ t r λγ r + φ r! λ r Γφ r + φ + λ Γ + φ λ + φ λ + φ r + φ λ + φγφ r r t r r! t r r! Γ r + φ + λ r Γφ + r + φ Γ r λ r Γφ + φ Corollary Proof For the radom varable T wth GWL dstrbuto, the r-th momet s gve by r + φ + λ Γ r + φ μ r E[T r ] λ + φλ r Γφ Note that, μ r M r X d M X dt ad the result follows 6 Corollary For the radom varable T wth GWL dstrbuto, the r-th cetral momet s gve by M r E[T μ] r r r r r μ r E[T ] + φ λλ + φγφ + φ + λ Γ r + φ + λ Γ + φ λ + φλ Γφ Corollary A radom varable T wth GWL dstrbuto has the mea ad varace respectvely gve by + φ + λ Γ + φ μ, λλ + φγφ λλ + φ + φ + λ Γ + φ Γ + φ + λ + φ σ λ λ + φ Γφ Proof From 6 ad cosderg r we have μ μ The secod result follows from cosderg r wth some algebrac mapulatos Aother momet fucto that ca be easly acheved for GWL dstrbuto ad plays a mportat role formato theory s gve by ψφ log λ +λ + φ E[logT] 8 9 Page of 8

5 Ramos & Louzada, Coget Mathematcs 6, : 56 Survval propertes I ths secto, we preset the survval, hazard, ad mea resdual lfe MRL fucto for the GWL dstrbuto The survval fucto of T s gve by St φ, λ, Γ[ φ, λt ] λ + φ+λt φ e λt λ + φγφ where Γx, y x wy e x dw s called upper complete gamma The hazard fucto s gve as ht φ, λ, f t φ, λ, St φ, λ, λ φ t φ λ +λt e λt Γ [ φ, λt ] λ + φ+λt φ e λt The behavor of the hazard fucto whe t ad t are, respectvely, gve by h, f φ < λ, λ + φγφ f φ, f φ > ad h Theorem 5 The hazard rate fucto ht of the GWL dstrbuto s creasg, decreasg, bathtub, umodal, or decreasg-creasg-decreasg shaped Proof The theorem proposed by Glaser 98 s ot easly appled the GLW dstrbuto Sce the hazard rate fucto s complex, we cosdered the followg cases: Let, the GWL dstrbuto reduces to the WL dstrbuto Ghtay et al proved that the hazard fucto s bathtub-shaped creasg f <φ<φ >, for all λ > Let φ, the GWL dstrbuto reduces to the PL dstrbuto Cosderg β λ, Ghtay et al proved that the hazard fucto s { } creasg whe <, β> ; } { decreasg whe { <, β> or <<, β } ; { decreasg-creasg-decreasg f <<, <β< } Let ad λ, from Glaser s theorem Glaser, 98, the hazard rate fucto s decreasg shaped umodal for <φ< φ >, f φ < λ, f φ, f φ > These propertes make the GWL dstrbuto a flexble model for relable data Fgure gves examples of the shapes of the hazard fucto for dfferet values of φ, λ ad Fgure Hazard fucto shapes for GWL dstrbuto ad cosderg dfferet values of φ, λ ad ht 6 8 φ, λ5, φ5, λ5, 5 φ, λ5, φ, λ5, 5 φ5, λ, ht 5 6 φ, λ5, φ5, λ5, 6 φ, λ5, 8 φ5, λ5, 5 φ, λ5, t t Page 5 of 8

6 Ramos & Louzada, Coget Mathematcs 6, : 56 The MRL has bee wdely used survval aalyss ad represets the expected addtoal lfetme gve that a compoet has survved utl tme t Proposto 6 rt φ, λ, The MRL fucto rt φ, λ, of the GWL dstrbuto s gve by φ + + λ Γ φ +, λt λtλ + φγ φ, λt λ[λ + φγφ, λt +λt φ e λt ] Proof Note that rt φ, λ, yf y λ, φ dy t St t St p yf y λ, φ dy + p yf y λ, φ dy t x t φ + + λ Γ φ +, λt λtλ + φγ φ, λt λ[λ + φγφ, λt +λt φ e λt ] The behavor of the MRL fucto whe t ad t are, respectvely, gve by r λλ + φγφ Etropy I formato theory, etropy has played a cetral role as a measure of ucertaty assocated wth a radom varable Shao s etropy s oe of the most mportat metrcs formato theory For the GWL dstrbuto, Shao s etropy ca be obtaed by solvg H S φ, λ, ad r, f < λ, f, f > λ φ t φ λ +λt e λt log λ + φγφ f t φ, λ, dt Proposto A radom varable T wth GWL dstrbuto has Shao s etropy gve by H S φ, λ, logλ + φ+log Γφ log log λ where ηφ, λ Proof ψφφ From the Equato, we have λ + y logλ + yy φ e y dy φ + φ + λ λ + φ φ λ + φ ηφ, λ λ + φγφ λ log u logλ log u log u φ du H S φ, λ, log φ log λ + logλ + φ+logγφ + λ E[T ] φ E[log T] E [ logλ +λt ] 5 6 Note that E [ logλ +λt ] logλ +λt λ φ t φ λ +λt e λt λ + φγφ dt, usg the chage of varable y λt ad after some algebra Page 6 of 8

7 Ramos & Louzada, Coget Mathematcs 6, : 56 E [ logλ +λt ] λ + y logλ + yy φ e y dy λ + φγφ ηφ, λ λ + φγφ From Equatos 6 ad, we ca easly fd the soluto of E[T ] ad E[log T] ad the result as follows Aother popular etropy measure s proposed by Rey 96 Some recet applcatos of the Rey etropy ca be see Popescu ad Aordachoae If T has the probablty desty fucto the Rey etropy s defed by ρ log f ρ x dx Proposto 8 A radom varable T wth GWL dstrbuto, has the Rey etropy gve by H R ρ ρ log + log λ ρ logλ + φ+log Γφ logδρ, φ, λ, ρ 8 where δρ, φ, λ, y ρφ ρ+ λ + y ρ e ρy dy Proof The Rey etropy s gve by H R ρ ρ log ρ λ ρ λ + φ ρ Γφ ρ λt ρ ρ log ρ λ ρ λ + φ ρ Γφ ρ ρ log y φ ρφ ρ+ ρ λ ρ λ + φ ρ ρ δρ, φ, λ, Γφ λ +λt ρ e ρλt dt λ + y ρ e ρy dy ad wth some algebra the proof s completed Lorez curves The Lorez curve Boferro, 9 s a well-kow measure used relablty, come equalty, lfe testg ad reewal theory The Lorez curve for a o-egatve T radom varable s gve through the cosecutve plot of LFt t xf x dx xf x dx μ t xf x dx Proposto 9 The Lorez curve for the GWL dstrbuto s Lp + φ + λ γ where F p t p φ +, λf p λf p φ e [ ] + φ + λ Γ + φ λf p Page of 8

8 Ramos & Louzada, Coget Mathematcs 6, : 56 Methods of estmato I ths secto, we preset eght dfferet estmato methods for the parameters φ, λ ad of the GWL dstrbuto Maxmum lkelhood estmato The maxmum lkelhood method has bee wdely used due to ts better asymptotc propertes The estmates are obtaed by maxmzg the lkelhood fucto Let T,, T be a radom sample where T GWLφ, λ,, the lkelhood fucto s gve by { } { λ φ Lφ, λ, ;t t φ λ + φγφ λ +λt exp λ The log-lkelhood fucto lφ, λ, ;t log Lφ, λ, ;t s gve by lφ, λ, ;t log + φ log λ logλ + φ log Γφ+φ From the expressos φ are log λ+ logt φ λ ad + + log λ +λt λ + λ t λ +t + φ log λ+ φ t lφ, λ, ;t, lφ, λ, ;t, lφ, λ, ;t, the lkelhood equatos λ λ + φ + ψ φ λ t + λ + φ λt logt + log λt λ λ + λt where ψk log Γk Γ k Numercal methods such as Newto-Rapsho are requred to fd k Γk the soluto of the olear system Note that from ad ad after some algebra we have MLE λ + φ + ψ φ log λ+ logt φ MLE λ t log λt log λ+ λt log λt ṋ λ+ λt logt Uder mld codtos, the maxmum lkelhood estmates MLEs are asymptotcally ormal dstrbuted wth a jot multvarate ormal dstrbuto gve by φ MLE, λ MLE, MLE N [ φ, λ,, I φ, λ, ] as t log λt, t } logt 9 5 where Iφ, λ, s the Fsher formato matrx s gve as Iφ, λ, I φ,φ φ, λ, I φ,λ φ, λ, I φ, φ, λ, I φ,λ φ, λ, I λ,λ φ, λ, I λ, φ, λ, I φ, φ, λ, I λ, φ, λ, I, φ, λ,, 6 ad the elemets of the matrx are gve Appedx Page 8 of 8

9 Ramos & Louzada, Coget Mathematcs 6, : 56 Momets estmators The method of momets s oe of the oldest methods used for estmatg parameters statstcal models The momets estmators MEs of the GLW dstrbuto ca be obtaed by equatg the frst three sample momets x t, t ad t wth the theoretcal momets + φ + λ Γ + φ + φ + λ Γ + φ t t λ + φλγφ λ + φλ Γφ ad + φ + λ Γ + φ t λ + φλ Γφ Therefore, the ME φ ME, λ ME ad ME, ca be obtaed by solvg the o-lear equatos j + φ + λ Γ j + φ t j, j,, λ + φλ j Γφ Ordary ad weghted least-square estmate Let t, t,, t be the order statstcs the same otato s assumed for the ext subsectos of the radom sample of sze from Ft φ, λ, The least square estmators φ LSE, λ LSE ad LSE ca be obtaed by mmzg wth respect to φ, λ ad Equvaletly, the estmates ca be obtaed by solvg the o-lear equatos where Vφ, λ, [ F t φ, λ, ] + [ F t φ, λ, ] Δ + j t φ, λ,, j,, Δ t φ, λ, φ F t φ, λ,, Δ t φ, λ, λ F t φ, λ, ad Δ t φ, λ, F t φ, λ, Note that the soluto of Δ for,, volves partal dervatves of the lower complete gamma fucto However, ths ca be easly acheved umercally wth hgh precso The weghted least-squares estmates WLSEs, φ WLSE, λ WLSE ad WLSE, ca be obtaed by mmzg Wφ, λ, [ F t φ, λ, ] + These estmates ca also be obtaed by solvg the o-lear equatos [ F t φ, λ, ] Δ + j t φ, λ,, j,,, where Δ φ, λ,, Δ φ, λ, ad Δ φ, λ, are gve Page 9 of 8

10 Ramos & Louzada, Coget Mathematcs 6, : 56 Method of maxmum product of spacgs The MPS method s a powerful alteratve to MLE for the estmato of ukow parameters of cotuous uvarate dstrbutos Proposed by Cheg ad Am 99, 98, ths method was also depedetly developed by Raeby 98 as a approxmato to the Kullback Lebler formato measure Cheg ad Am 98 proved desrable propertes of the MPS such as asymptotc effcecy, varace, ad more mportatly, the cosstecy of maxmum product of spacg estmators holds uder more geeral codtos tha for MLEs Let D φ, λ, F t φ, λ, F t φ, λ,, for,,, +, be the uform spacgs of a radom sample from the GWL dstrbuto, where Ft φ, λ, ad Ft + φ, λ, Clearly + D φ, λ, The MPS estmates φ MPS, λ MPS ad MPS are obtaed by maxmzg the geometrc mea of the spacgs Gφ, λ, [ + D φ, λ, ] + wth respect to φ, λ ad, or, equvaletly, by maxmzg the logarthm of the geometrc mea of sample spacgs 8 Hφ, λ, + log D + φ, λ, 9 The estmates φ MPS, λ MPS ad MPS of the parameters φ, λ ad ca be obtaed by solvg the olear equatos + [ ] Δ + D φ, λ, j t φ, λ, Δ j t φ, λ,, j,,, where Δ φ, λ,, Δ φ, λ, ad Δ φ, λ, are gve respectvely Note that f t +k t +k t the D +k φ, λ, D +k φ, λ, D φ, λ, Therefore, the MPS estmators are sestve to closely spaced observatos, especally tes Whe the tes are due to multple observatos, D φ, λ, should be replaced by the correspodg lkelhood f t, φ, λ, sce t t Uder mld codtos for the GWL dstrbuto, the MPS estmators are asymptotcally ormal dstrbuted wth a jot trvarate ormal dstrbuto gve by φ MPS, λ MPS, MPS N [ φ, λ,, I φ, λ, ] as 5 The Cramer-vo Mses mmum dstace estmators The Cramer-vo Mses estmator s a type of mmum dstace estmators also called maxmum goodess-of-ft estmators ad s based o the dfferece betwee the estmate of the cumulatve dstrbuto fucto ad the emprcal dstrbuto fucto Luceño, 6 Macdoald 9 motvated the choce of the CME estmators provdg emprcal evdece that the bas of the estmator s smaller tha the other mmum dstace estmators The Cramer-vo Mses estmates φ CME, λ CME ad CME of the parameters φ, λ ad are obtaed by mmzg Cφ, λ, + F t φ, λ,, wth respect to φ, λ ad These estmates ca also be obtaed by solvg the olear equatos: Page of 8

11 Ramos & Louzada, Coget Mathematcs 6, : 56 F t φ, λ, Δ j t φ, λ,, j,,, where Δ φ, λ,, Δ φ, λ, ad Δ φ, λ, are gve respectvely 6 The Aderso Darlg ad Rght-tal Aderso Darlg estmators Aother type of mmum dstace estmator s based o ADE statstc ad s kow as ADE estmator The ADE estmates φ ADE, λ ADE ad ADE of the parameters φ, λ ad are obtaed by mmzg, wth respect to φ, λ ad, the fucto Aφ, λ, log F t φ, λ, + log S t + φ, λ, These estmates ca also be obtaed by solvg the olear equatos [ Δ j t φ, λ, F t φ, λ, Δ j t+ φ, λ, ] S t + φ, λ,, j,, The Rght-tal ADE estmates φ RADE, λ RADE ad RADE of the parameters φ, λ ad are obtaed by mmzg the fucto Rφ, λ, F t : φ, λ, log S t+ : φ, λ, wth respect to φ, λ ad These estmates ca also be obtaed by solvg the olear equatos: Δ j t: φ, λ, + Δ j t+ : φ, λ, S, j,, t + : φ, λ, where Δ φ, λ,, Δ φ, λ, ad Δ φ, λ, are gve respectvely Smulato study I ths secto, a tesve smulato study s preseted to compare the effcecy of the estmato procedures for parameters of the GWL dstrbuto The followg procedure was adopted: Geerate pseudo-radom values from the GWLφ, λ, wth sze Usg the values obtaed step, calculate φ, λ ad va -MLE, -MPS, -ADE, -RTADE, 5-LSE, 6-WLSE, -ME, 8-CME Repeat the steps ad N tmes Usg θ φ, λ, ad θ φ, λ,, compute the mea relatve estmates MRE θ N,j θ j N ad the mea square errors MSE N θ,j θ j, for,, N Cosderg ths approach, the most effcet estmato method wll have MREs closer to oe ad MSEs closer to zero The results were computed usg the software R usg the seed 5 to geerate the pseudo-radom values The tal values cosdered were the same values used to geerate the radom samples The chose values to perform ths procedure were N, ad 5, 6,, 5 For reasos of space, we have preseted the results oly for θ, 5, However, the followg results are smlar for other choces of θ Page of 8

12 Ramos & Louzada, Coget Mathematcs 6, : 56 For ths comparso to be meagful, the estmato procedures eed to be performed uder same codtos However, for some partcular samples ad estmato methods, the umercal techques do ot work well fdg the parameter estmates Therefore, a rate study s preseted to verfy the frequecy of covergece of the umercal solutos Ths procedure s carred out by coutg the umber of tmes each estmato fals fdg the umercal soluto I Fgure we preset the proporto of falure from each method From Fgure, the MLE, LSE, WLSE, ME, ad the CME estmators fal fdg the parameter estmates for a sgfcat umber of samples Therefore, such methods are ot recommeded for estmato of the GLW parameters Hereafter, we cosder the MPS, ADE, RADE estmators due to ther better computatoal stablty The MLE s cosdered oly for llustratve purposes sce t s the most used estmato method Fgure presets the MREs, MSEs for the estmates of φ, λ ad usg the MLE, MPS, ADE, RADE wth N smulated samples ad dfferet values of θ, 5, ad The horzotal les both fgures correspod to MREs ad MSEs beg oe ad zero, respectvely From these results, the MSE of the MLE, MPS, ADE, ad RADE estmators ted to zero for large ad also, as expected, the values of MREs ted to oe, e the estmates are cosstet ad asymptotcally ubased for the parameters For small sample szes, the MLE has the largest MSEs The MPS has smaller MSEs wth MREs closer to oe for almost all values of Addtoally, the MPS, ADE, ad RADE estmators were the oly methods that were able to fd φ, λ ad for all the 6 geerated samples Therefore, combg all results wth the good propertes of the MPS method such as cosstecy, asymptotc effcecy, ormalty ad varace, we coclude that the MPS estmators are a hghly compettve method compared to the maxmum lkelhood for estmatg the parameters of the GWL dstrbuto Fgure Proporto of falure from N smulated samples, cosderg dfferet values of usg the followg estmato method -MLE, -MPS, -ADE, -RTADE, 5-LSE, 6-WLSE, -ME, 8-CME Proporto Page of 8

13 Ramos & Louzada, Coget Mathematcs 6, : 56 Fgure MREs, MSEs related from the estmates of φ 5, λ ad 5 for N smulated samples, cosderg dfferet values of obtaed usg the followg estmato method -MLE, -MPS, -ADE, -RTADE MRE φ MSE φ MRE λ 5 MSE λ MRE 5 5 MSE Applcato I ths secto, we compare the GWL dstrbuto wth other three-parameter lfetme dstrbutos cosderg two data-sets, the frst wth bathtub hazard rate ad the other wth the creasg hazard fucto The followg lfetme dstrbutos were cosdered The GG dstrbuto wth pdf gve by f t Γφ β φ t φ e βt where β >, φ> ad > The GW dstrbuto where the pdf s f t φ t φ λt φ λ where λ R, φ> ad > The GEP dstrbuto wth pdf gve by f t βφ e φ e φ βt+φ exp βt e φ+φ exp βt where β >, φ> ad > The EW dstrbuto wth pdf Page of 8

14 Ramos & Louzada, Coget Mathematcs 6, : 56 f t φβ t β exp t β exp t β φ where β >, φ> ad > The TTT-plot total tme o test s cosdered order to verfy the behavor of the emprcal hazard fucto Barlow & Campo, 95 The TTT-plot s obtaed through the plot of [r /, Gr / ] where r Gr t + rt r t, r,,,,, ad t s the statstcal order If the curve s cocave covex, the hazard fucto s creasg decreasg O the other had, whe t starts covex ad the becomes cocave cocave ad the covex the hazard fucto has bathtub verse bathtub shape The goodess of ft s checked cosderg the Kolmogorov Smrov KS test Ths procedure s based o the KS statstc D sup F t Ft;φ, λ,, where sup t s the supremum of the set of dstaces, F t s the emprcal dstrbuto fucto ad Ft;, β, λ s cdf A hypothess test s coducted at the 5% level of sgfcace to test whether or ot the data come from Ft;, β, λ I ths case, the ull hypothess s rejected f the retured p-value s smaller tha 5 To carry out the model selecto, the followg dscrmato crtero methods are adopted: AIC Akake formato crtera ad AICc Corrected Akake formato crtero computed, respectvely, by AIC l θ;t+k ad AICc AIC + k k + k, where k s the umber of parameters to be ftted ad θ s estmato of θ For a set of caddate models for t, the best oe provdes the mmum values 5 Lfetmes data Aarset 98 presets the data-set see Table related to the lfetme hours of 5 devces o test Fgure 5 shows left pael the TTT-plot, mddle pael the ftted survval supermposed to the emprcal survval fucto ad rght paels the hazard fucto adjusted by GWL dstrbuto Table Lfetmes data hours related to a devce o test Fgure 5 left pael the TTTplot, mddle pael the ftted survval supermposed to the emprcal survval fucto ad rght paels the hazard fucto adjusted by GWL dstrbuto Gr/ 6 8 St 6 8 Emprcal Ge WL Ge Gamma Ge Webull Exp Webull Ge EP ht r/ Tempos Tempos Page of 8

15 Ramos & Louzada, Coget Mathematcs 6, : 56 Table Results of AIC ad AICc crtera ad the p-value from the KS test for all ftted dstrbutos cosderg the Aarset dataset Crtera Ge WL Ge Gamma Ge Webull Exp Webull Ge EP AIC AICc KS Fgure 6 left pael the TTTplot, mddle pael the ftted survval supermposed to the emprcal survval fucto ad rght paels the hazard fucto adjusted by GWL dstrbuto Gr/ 6 8 St 6 8 Emprcal Ge WL Ge Gamma Ge Webull Exp Webull Ge EP ht r/ m / m / Table presets the AIC ad AICc crtera ad the p-value from the KS test for all ftted dstrbutos cosderg the Aarset dataset Comparg the emprcal survval fucto wth the adjusted dstrbutos, t ca be observed that the GWL dstrbuto s as a better ft Ths result s also cofrmed from the AIC ad AICC see Table sce GWL dstrbuto has the mmum values ad also the p-values retured from the KS test are greater tha 5 It should be emphaszed that cosderg a sgfcace level of 5%, the others models are ot able to ft the proposed data Table dsplays the MPS estmates, stadard errors, ad the cofdece tervals CI for φ, λ ad of the GWL dstrbuto 5 Average flows data The study of average flows has bee proved to be of hgh mportace to protect ad mata aquatc resources streams ad rvers Reser, Wesche, & Estes, 989 I ths secto, we cosder a real data-set related to the average flows m /s of the Catarera system durg Jauary at São Paulo cty Brazl It s worth metog that the Catarera system provdes water to 9 mllo people the São Paulo metropolta area The data-set avalable Table was obtaed from the Natoal Water Agecy from 9 to Table MPS estmates, stadard-error ad 95% CI for φ, λ ad θ θ MPS SE θ CI 95% θ φ 5 9 9; 5 λ 8 ; ; 958 Page 5 of 8

16 Ramos & Louzada, Coget Mathematcs 6, : 56 Table Jauary average flows m /s of the Catarera system Table 5 Results of AIC ad AICc crtera ad the p-value from the KS test for all ftted dstrbutos cosderg the data-set related to the jauary average flows m /s of the Catarera system Crtera Ge WL Ge Gamma Ge Webull Exp Webull Ge EP AIC AICc KS Table 6 ML estmates, stadard-error ad 95% CI for φ, λ ad θ θ MLE SE θ CI 95% θ φ ; λ 55 8; ; 989 I ths secto, we cosder the ML estmator showg that both MPS or MLE could be used successfully applcatos Fgure 6 shows left pael the TTT-plot, mddle pael the ftted survval supermposed to the emprcal survval fucto ad rght paels the hazard fucto adjusted by GWL dstrbuto Table 5 presets the AIC ad AICc crtera ad the p-value from the KS test for all ftted dstrbutos cosderg the data-set related to the Jauary average flows m /s of the Catarera system From the emprcal survval fucto ad the adjusted dstrbutos, t ca be observed that the GWL dstrbuto s better Ths result s also cofrmed from AIC ad AICC sce GWL dstrbuto has the mmum values ad the p-values retured from the KS test are greater tha 5 Table 6 dsplays the ML estmates, stadard errors, ad the CI for φ, λ ad of the GWL dstrbuto 6 Cocludg remarks To summarze, we have proposed a three-parameter lfetme dstrbuto The GLW dstrbuto s a straghtforward geeralzato of the WL dstrbuto proposed by Ghtay et al, whch accommodates creasg, decreasg, decreasg-creasg-decreasg, bathtub, ad umodal hazard rate makg the GWL dstrbuto a flexble model for relable data The mathematcal propertes of ths dstrbuto are also dscussed The estmato procedures for the parameters of GWL dstrbuto are also derved cosderg eght estmato methods Sce t s ot feasble to compare these methods theoretcally, we have preseted a extesve smulato study order to detfy the most effcet procedure We Page 6 of 8

17 Ramos & Louzada, Coget Mathematcs 6, : 56 observed that the MLE, ME, LSE, WLSE, ad the CME estmators fal fdg the parameter estmates for a sgfcat umber of samples The smulatos showed that the MPS maxmum product of spacg s the most effcet method for estmatg the parameters of the GWL dstrbuto comparso to ts compettors Fally, two data-sets were aalyzed for llustratve purposes provg that the GWL dstrbuto outperforms several usual three parameter lfetme dstrbutos Ackowledgemets We are grateful to the Edtoral Board ad the revewers for ther valuable commets ad suggestos whch has mproved the mauscrpt Fudg The research was partally supported by CNPq, FAPESP, ad CAPES of Brazl Author detals PL Ramos E-mal: pedrolramos@uspbr F Louzada E-mal: louzada@cmcuspbr Isttute of Mathematcal Scece ad Computg, Uversdade de São Paulo, Sao Carlos-SP, Brazl Ctato formato Cte ths artcle as: The geeralzed weghted Ldley dstrbuto: Propertes, estmato, ad applcatos, PL Ramos & F Louzada, Coget Mathematcs 6, : 56 Refereces Aarset, M V 98 How to detfy a bathtub hazard rate IEEE Trasactos o Relablty, 6, 6 8 Al, S 5 O the bayesa estmato of the weghted ldley dstrbuto Joural of Statstcal Computato ad Smulato, 85, Al-Mutar, D, Ghtay, M, & Kudu, D 5 Ifereces o stress-stregth relablty from weghted ldley dstrbutos Commucatos Statstcs-Theory ad Methods,, 96 Bakouch, H S, Al-Zahra, B M, Al-Shomra, A A, March, V A, & Louzada, F A exteded ldley dstrbuto Joural of the Korea Statstcal Socety,, 5 85 Barlow, R E, & Campo, R A 95 Total tme o test processes ad applcatos to falure data aalyss Techcal report Berkeley, CA: DTIC Documet Barreto-Souza, W, & Bakouch, H S A ew lfetme model wth decreasg falure rate Statstcs,, 65 6 Barreto-Souza, W, & Crbar-Neto, F 9 A geeralzato of the expoetal-posso dstrbuto Statstcs & Probablty Letters, 9, 9 5 Boferro, C 9 Elemet d statstca geerale Freze: Seeber Cheg, R & Am, N 99 Maxmum product of spacgs estmato wth applcato to the logormal dstrbuto Mathematcal Report 9- Cardff: Uversty of Wales IST Cheg, R, & Am, N 98 Estmatg parameters cotuous uvarate dstrbutos wth a shfted org Joural of the Royal Statstcal Socety Seres B Methodologcal, 5, 9 Ghtay, M, Al-Mutar, D, Balakrsha, N, & Al-Eez, L Power ldley dstrbuto ad assocated ferece Computatoal Statstcs & Data Aalyss, 6, Ghtay, M, Alqallaf, F, Al-Mutar, D, & Husa, H A two-parameter weghted ldley dstrbuto ad ts applcatos to survval data Mathematcs ad Computers Smulato, 8, 9 Ghtay, M, Ateh, B, & Nadarajah, S 8 Ldley dstrbuto ad ts applcato Mathematcs ad Computers Smulato, 8, 9 56 Glaser, R E 98 Bathtub ad related falure rate characterzatos Joural of the Amerca Statstcal Assocato, 5, 66 6 Luceño, A 6 Fttg the geeralzed pareto dstrbuto to data usg maxmum goodess-of-ft estmators Computatoal Statstcs & Data Aalyss, 5, 9 9 Macdoald, P 9 A estmato procedure for mxtures of dstrbuto Joural of the Royal Statstcal Socety Seres B Methodologcal,, 6 9 Mazuchel, J, Louzada, F, & Ghtay, M Comparso of estmato methods for the parameters of the weghted ldley dstrbuto Appled Mathematcs ad Computato,, 6 Mudholkar, G S, Srvastava, D K, & Fremer, M 995 The expoetated webull famly: A reaalyss of the busmotor-falure data Techometrcs,, 6 5 Mudholkar, G S, Srvastava, D K, & Kolla, G D 996 A geeralzato of the webull dstrbuto wth applcato to the aalyss of survval data Joural of the Amerca Statstcal Assocato, 9, Popescu, T D, & Aordachoae, D Sgal segmetato tme-frequecy plae usg rey etropy-applcato sesmc sgal processg I Coferece o Cotrol ad Fault-Tolerat Systems SysTol pp Nce: IEEE Raeby, B 98 The maxmum spacg method A estmato method related to the maxmum lkelhood method Scadava Joural of Statstcs,, 9 Reser, D W, Wesche, T A, & Estes, C 989 Status of stream flow legslato ad practces orth amerca Fsheres,, 9 Rey, A 96 O measures of etropy ad formato I Fourth Berkeley Symposum o Mathematcal Statstcs ad Probablty,, 56 Stacy, E W 96 A geeralzato of the gamma dstrbuto The Aals of Mathematcal Statstcs,, 8 9 Torab, H, Falahat-Nae, M, & Motazer, N A exteded geeralzed ldley dstrbuto ad ts applcatos to lfetme data Joural of Statstcal Research of Ira,, Wag, M, & Wag, W press Bas-corrected maxmum lkelhood estmato of the parameters of the weghted ldley dstrbuto Commucatos Statstcs- Smulato ad Computato, 6, 5 55 Zakerzadeh, H, & Dolat, A 9 Geeralzed ldley dstrbuto Joural of Mathe-matcal Exteso,, Page of 8

18 Ramos & Louzada, Coget Mathematcs 6, : 56 Appedx The elemets of the Fsher formato matrx are [ ] lθ;t I φ,φ E φ λ + φ + ψ θ [ ] lθ;t I φ,λ E φ λ λ + λ + φ [ ] lθ;t logλ ψφ+logλ λ + φ I φ, E φ [ ] lθ;t I λ,λ E φ λ λ + λ ψφ logλ+λ + φ [ T λ λ λt ] + E λ +λt λ + φ [ ] lθ;t φλ + φ + ψφ + ψφ I, E + λ + φ [ ] λ + φ + ψφ+ λλt logλt + E λ + φ λ +λt [ ] lθ;t I,λ E φ λ + φψφ + φ +φ + ψφ + + λ λ λλ + φ [ + λ T ] λt logλt φ + λ + Γ φ + E + λ +λt λ + φγφ [ λ T logλt+λt E λ +λt ] 6 The Authors Ths ope access artcle s dstrbuted uder a Creatve Commos Attrbuto CC-BY lcese You are free to: Share copy ad redstrbute the materal ay medum or format Adapt remx, trasform, ad buld upo the materal for ay purpose, eve commercally The lcesor caot revoke these freedoms as log as you follow the lcese terms Uder the followg terms: Attrbuto You must gve approprate credt, provde a lk to the lcese, ad dcate f chages were made You may do so ay reasoable maer, but ot ay way that suggests the lcesor edorses you or your use No addtoal restrctos You may ot apply legal terms or techologcal measures that legally restrct others from dog aythg the lcese permts Page 8 of 8