HAZARD RATE ESTIMATION FOR LOCATION-SCALE DISTRIBUTIONS UNDER COMPLETE AND CENSORED DATA

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1 HAZARD RATE ESTIMATION FOR LOCATION-SCALE DISTRIBUTIONS UNDER COMPLETE AND CENSORED DATA Bas Suucu Depame o Sascs Mddle Eas Techcal Uvesy Akaa Tukey. Ioduco I elably ad le esg sudes, he opc o esmag hazad ae has eceved gea aeo ece yeas sce a esmae o hazad ae s a que useul ool o makg decsos. Some woks have cluded opaamec appoaches whle some have cosdeed paamec sucual models o complee as well as cesoed daa ses; see Meeke e al. 99, Aoads ad Gégoe 999, Ra ad Sgh 003, Bezady e al. 005, Buel ad Come 008, ad Mahapaa e al. 0. Depedg o he shapes o he hazad ae, ececes de makedly acoss poposed esmaos. Ths suao s emakable especally whe dee esmao echques ae ulzed o ukow paamees o udelyg dsbuos paamec appoaches. Tha s, esmaed hazad ae ad also elably a a specc me po as ucos o hese esmaos leads o cosse coveage pobables as dsbuoal covegece o hazad ae esmao may o be a he desed ae o cea sample szes. Ths mauscp ocuses o he esmao o moooe hazad ae whe he dsbuo o coce s o locao-scale ype. A vey smple ad ece appomao o hazad ae o a complee sample s oduced as a uco o ode sascs, whch allows a as covegece o he asympoc dsbuo whle achevg hghly accuae coveage pobables o codece evals CI. Ececy o he mehod s demosaed o smulao sudes by cosdeg a umbe o locao-scale dsbuos havg vaous hazad shapes. We also cosde he case o cesoed samples by copoag he pedced uue ode sascs o he pcue. The emade o he pape s ogazed as ollows: I Chape, we dscuss he esmao o ukow paamees o locao-scale amles ad he ececes. I Chape 3, we popose a lea appomao o he hazad ae uco o esmae he hazad ae a me o dee udelyg locao-scale dsbuos. We dscuss he asympoc popees o he hazad ae esmao ad pese he ececes o he ew mehod by compag wh hose o classcal appoach Chape 4. Fo he cesoed samples, ecessy o pedcg uue ode sascs he ovel mehod s eplaed Chape 5. Moeove, a secod appoach based o epeced spacgs s oduced o acheve he same goal. I he same chape, some gaphcal dsplays ae povded o llumae he dscusso o boudaes o CI s o hazad ae beoe cocludg.

2 . Paamee Esmao Le us have a locao-scale dsbuo havg he pobably desy uco X, ; whee ad ae locao ad scale paamees, especvely. Fo, he hazad ae a me po ca be we as F h whee z ad z F ae pobably desy uco ad dsbuo uco o he sadadzed adom vaable z, especvely. To esmae he hazad ae, we sly eed o esmae he ukow locao ad scale paamees. Thoughou he wok, we wll cosde wo ypes o esmao echques; leas squaes esmao ad moded mamum lkelhood esmao o Tku 967. esmaos ae well-kow ad we do o gve he deals abou hem hs mauscp. To bely oduce, we he loglkelhood uco o a cesoed sample - lages obsevaos cesoed as l l l l F L. 3 Realze ha whe, we have he complee sample o sze. The, he paal devaves wh espec o he ukow locao ad scale paamees ca be we as l F L 4 ad l F L. 5

3 Noe also ha sample obsevaos 4 ad 5 ca be eplaced by odeed sascs as he complee sums ae vaa o odeg. The, dee g z z z ad g z 3 z F z g z z z,, whee z s he h sadadzed ode sasc. Depedg o he sucual om o he pd o he adom vaable X, he ucos g, g ad g 3 ae mosly olea ad he basc cause o havg eplc soluos o he lkelhood ucos. To ovecome hs poblem, Tku 967 poposed a modcao o he mamum lkelhood appoach, whch leaded o vey ece eplc soluos o he ukow locao ad scale paamees o udelyg dsbuos. I ac, he esmaos ae obus ad asympocally ully ece; see also Tku ad Suesh 99. Le oe o hese olea ucos be g z. The, a Taylo epaso aoud he epeced value o he h sadadzed ode sasc m gves us he lea appomao g z α β z 6 whee α ad β ae he coespodg cosas obaed om he mehod; see Tku 967 ad Tku ad Suesh 99. By eplacg all olea ucos wh he appomae lea oms, we oba eplc soluos o he equaos 4 ad 5, whch lead o Tku's esmaos. I ac, he esmaos ae asympocally equvale o ML esmaos ad ae vey hghly ece. Specc oms o he esmaos ad he dsbuoal popees wll be eamed he e secos whle wokg wh dee dsbuos. 3. Esmag Hazad Rae The hazad ae a me o a locao-scale dsbuo ca be deed as δ F h δ δ 7 whee δ. Sce he paamees ae ukow he eal le, we s esmae hem by usg a pope esmao echque ad plug hem he hazad uco 7 o esmae he hazad ae. Howeve, hs bgs abou some dsbuoal poblems o he esmao h ˆ δ. Tha 3

4 s, he asympoc dsbuo o h ˆ δ cao be obaed easly ad, heeoe, s que dcul o cosuc a CI wh a eleva coveage pobably. A appomao echque smla o ha dscussed Seco helps us o ovecome hese dsbuoal poblems. The ew lea appomao mehod does o oly solve he dsbuoal suggle, bu also povdes a vey smple ad paccal sucual om wh vey hgh ececy. To epla he mehod, cosde he ollowg lea appomao. ˆ ˆ F α β ˆ δ 8 ˆ ˆ whee ˆ ˆ δ. Deely om Seco, δ s o a sadadzed ode sasc whch ca ˆ be epaded aoud s epecao by Taylo sees appoach. O he ohe had, s always bewee wo cosecuve ode sascs o eample k ad k. Theeoe, he epeced values o he sadadzed ode sascs zk ad z k ca be cosdeed as he boudaes o a ew leazao o h ˆ δ. The, he esmaed hazad ae a po ca be we as whee k ˆ h δ α β ˆ δ 9 α h m k β m k h mk h mk β ; 0 m m k k m beg he epeced value o he kh sadadzed ode sasc z k. Asympoc Dsbuo o he Hazad Rae Esmao: Due o he popees o ˆ esmaos, he asympoc dsbuo o m s omal wh mea m ad ˆ vaace, whee V ˆ m ; m s a cosa akg dee values o dee ˆ dsbuos. Accodgly, h ˆ δ α β s asympocally omally dsbued wh ˆ mea α β ad vaace β m. Fom hee, s vey easy o cosuc α % CI s o boh ad esmaos. Fo moooe hazad sucues, he CI s ae gve as ollows: 4

5 : E: ˆ β ˆ β P α β zα h δ α β zα α, β > 0 ˆ m ˆ m β β P α β zα h δ α β zα α, β > 0 ; s s ad s beg he leas squaes esmaos o ad, especvely. Remak: I geeal, he -eval wll o he aveage be shoe ha he E-eval sce m s geae ha. 4. Hazad Rae Esmao o Seveal Dsbuos I hs seco, we wll wok o he hazad ae esmao pocess o sho-aled ad log-aled symmec dsbuos whch have moooe hazad aes. Peomaces o he esmaos wll be aalyzed hough smulao sudes. Fo he compaso pupose, wo basc ypes o esmao pocedues wll be cosdeed. The s oe s he hazad ae esmao obaed hough eplacg ukow paamees wh he esmaos o he hazad ae uco. The secod oe s due o he ew lea appomao echque whch ulzes wo cosecuve ode sascs. The esmaos obaed om he s appoach wll be amed as ad ad. wheeas hose obaed om he secod appoach wll be amed as 4.. Sho-Taled Symmec Dsbuo pd Sho-aled symmec STS dsbuo oduced by Tku ad Vaugha 999 has he whee { z }, < < z y C ep z π z y ; beg a ege,, > d d, 3 ad C j. 4 j j! j 0 j j! 5

6 6 The STS dsbuo s umodal o d <0. Fo moe deals abou he STS amly, see Tku ad Vaugha 999. The esmaos o a complee sample o sze o he STS dsbuo ae gve as Tku ad Akkaya, 004, p.68 m y ˆ β, m β, 5 4 ˆ C B B, 6 whee y B α, ˆ β m y k p C, [ ] 3 α ad [ ] β. 7 Whe, all β ae posve ad he equaos 7 ca be used. Whe >, howeve, some β coeces ake egave values. To ovecome he coveece he esmao sep, α ad β ae eplaced by α ad β ; { } 3 α ad { } β ; 8 The α % CI o he hazad ae a me ca be cosuced as he equao 0 wh 4 3 h h h h m ; d h, < d ; 9 see also Akkaya ad Tku 008 o moe deals. Sce he eac epeced values o he sadadzed ode sascs ae o avalable o he STS dsbuo, we use he smulaed

7 values o he lea appomao mehod. Calculaos o α ad β ae saghowad om 0. Leas squaes esmaos o ad ca be obaed as ad s, 0 whee { j }! j j j 0 j j! j j! j j 0 j j! s he vaace o he sho-aled symmec amly. I s aga vey easy o cosuc he α % CI o he hazad ae a me by smply ollowg he ules gve Seco 3. I Table, oe ca see he smulao esuls o he ececes o he hazad ae esmaos o he STS dsbuo. Fo cocseess, we oly gve he esuls o 0. The ececy behavos o ohe sample szes ad ohe paamee values ae almos he same. Esmaos based o he ae moe ece ha hose based o he. have almos he same ececy o he all quales. I Table, he coveage pobables o he CI s ae gve o ad ad. I s vey clea om he smulao esuls ha he lea appomao ulzg he esmaos s peomg vey much bee ha ha usg he esmaos. The epeced α 0.95 level s acheved he md-quales o he dsbuo. Towads he als o he STS dsbuo, he coveage pobably deceases o O he ohe had, he coveage pobably o alls below he level o 0.70 he lowe ad uppe als o he dsbuo. Oe ca also see he Fgue ad Fgue whch show he CI boudaes o he -evals o he same dsbuo. 7

8 Table : Meas ad Vaaces o Hazad Rae Esmaos o he Quales o he STS Dsbuo Q 0, 0,,, d 0 ˆ , ˆ. 006, , s Meas o 0 Vaaces o 0 Q Eac Table : Coveage Pobables α o he CI s o he Hazad Rae Esmaos o he STS Dsbuo 0, α 0.05, 0,,, d 0 ˆ , ˆ. 006, , s Q Coveage Pobables Q

9 Fgue : -eval o he quales o he STS dsbuo 0, α 0.05, 0,,, d 0 Fgue : -eval o he quales o he STS dsbuo 50, α 0.05, 0,,, d.0 9

10 4.. Log-Taled Symmec Dsbuo The pd o he log-aled symmec LTS dsbuo s gve by y y, p k p, < y < ; wh mea ad vaace ; k p -3 ad p see Tku ad Suesh, 99. The esmaos o a complee sample o sze ae obaed as eplaed Seco ad hey have he same sucues gve 5 ad 6 wh p B k α, y p C β y m ˆ 3 k whee k α, [ k ] k β ad [ k ] m p β k. 4 Epeced values o he sadadzed ode sascs ae avalable Tku ad Kuma 98 o p0.50, 0. Oe ca also see he coespodg epeced values o p.5, 0 Vaugha 99b ad hose o p Cauchy dsbuo, 6 Vaugha 994. Tku e. al. 000 povdes he aleave coeces α ad β case β < 0 o some ad hey ae gve by α ad { k } 0 β. 5 Table 3 ad 4 show he esuls o a smulao sudy o he ececes o he hazad ae esmaos obaed o he LTS dsbuo. As he case o he STS dsbuo, he same cocluso ca be daw ad he esmaos based o he esmao peom much moe ece ha hose based o he. As o he coveage pobables, esuls ae smla o hose obaed om he STS dsbuo. 0

11 Table 3: Meas ad Vaaces o Hazad Rae Esmaos o he Quales o he LTS Dsbuo Q 0, 0,, p 3 ˆ 0.009, ˆ. 049, , s Meas o 0 Vaaces o 0 Q Eac Table 4: Coveage Pobables α o he CI s o he Hazad Rae Esmaos o he LTS Dsbuo 0, 0,, p 3 ˆ 0.009, ˆ. 049, , s Q Coveage Pobables Q

12 4.3. Ohe Locao-Scale Dsbuos Smlaly o he STS ad he LTS dsbuos, we ca easly deve he hazad ae esmaos o ohe locao-scale dsbuos havg moooe hazad aes ad he CI's. esmaos o ukow paamees o hese dsbuos ae o dcul o oba due o he ece appomao o olea ems o lkelhood ucos. Oce esmaos ae obaed, s aga vey easy o esmae he hazad ae a specc po. Fo locao-scale dsbuos havg omoooc hazad aes, we ca sll make use o he same appoach. Ecep a he chage po, s possble o oba hghly accuae esmaes o hazad aes. As a uue sudy, he chage po poblem ca be woked ou moe deal ad ew appomao mehods ca be developed o cease accuaces. 5. Esmao Cesoed Samples Fo cesoed samples comg om locao-scale amles, ukow locao ad scale esmaos ca easly be obaed hough he appoach as eplaed Seco. Fo he hazad ae esmao, howeve, he pocedue poposed Seco 3 s o decly applcable sce he ode sascs he cesoed pas o samples cao be obseved. Fo a uue po, we cao obseve wo ode sascs ha wll be ulzed he lea appomao. Ths leads us o popose wo ew appoaches whch wll be helpul pedcg ode sascs o he leazao boudaes. Cosde a cesoed sample om a locao-scale amly ad assume las - obsevaos ae o obseved. I ode o leaze he hazad uco a po >, we eed o have a eval o wo cosecuve ode sascs coag. Sce hese ode sascs ae o avalable, we ca poceed wh he pedced values. To do ha, cosde he ollowg pedcve log-lkelhood uco o he cesoed sample ad he h ode sasc > : j L X F l,, F l l l 6 l F ; <. Takg he paal devaves wh espec o, ad gves olea ems as was dscussed Seco. Howeve, he appoach easly akes cae o he poblem ad povdes closed-om soluos o he esmaos ˆ, ˆ ad. See also Raqab 997 who also cosdeed he same poblem by usg he smla appomao echques. Oce he uue ˆ

13 ode sascs ae pedced, hazad ae esmao a po ca easly be hadled by smply deemg he value o k sasyg ˆ < ˆ. The, he esmao ollows om he pocedue eplaed Seco 3. k < k Aohe way o deemg he value o k s o make use o he geealzed spacgs o he locao-scale dsbuo sce he spacgs ae ubased esmaos o he scale paamee. Le ˆ ad ˆ be he esmaos obaed by he appoach o he cesoed sample, 3,...,. The, a esmao o a uue sasc s, ˆ j ˆ j j 7 whee s he epeced value o he h sadadzed ode sasc. Oe ca wok o he ececy o hs esmao as a uue sudy. Reeeces Akkaya, A. D. ad Tku, M. L., 008. Sho-aled Dsbuos ad Iles, Tes, 7, Aoads, A., Gegoe, G. 999, "Desy ad Hazad ae esmao o gh-cesoed daa usg wavele mehods," J. R. Sas. Soc. B, 6,, pp Bezady, P. H., Boey, G. E. ad Gaou, A Cosse esmao o he desy ad hazad ae ucos o cesoed daa va he wavele mehod. Sascs ad Pobably Lees, 744, Buel E, Come F 008 Adapve esmao o hazad ae wh cesoed daa. Comm Sas Theoy Mehods 378 0: Mahapaa, A. K., Kuma, S. ad Vellasamy, P. 0. Smulaeous esmao o hazad aes o seveal epoeal populaos, Sasca Neeledca, 66, -3. Meeke, W. Q., Escoba L. A., ad Hll, D. A. 99. Sample szes o esmag he Webull hazad uco om cesoed samples, IEEE Tasacos o Relably R-4, Ra, B. Ad Sgh, N Hazad ae esmao om complee ad uclea waay daa, Rel. Eg. Sys. Sa., 8, Raqab, M. Z Moded Mamum Lkelhood Pedcos o Fuue Ode Sascs om Nomal Samples, Compuaoal Sascs ad Daa Aalyss 5,

14 Tku, M.L Esmag he mea ad sadad devao om a cesoed omal sample. Bomeka, 54, Tku, ML. ad Akkaya, A.D Robus Esmao ad Hypohess Tesg, New Age Ieaoal Publshes P: New Delh, 004. Tku, M.L. ad Kuma, S. 98 Epeced values ad vaaces ad covaaces o ode sascs o a amly o symmecal dsbuos Sude's. Seleced Tables Mahemacal Sascs 8, Ameca Mahemacal Socey, Povdece, RI: Tku, M. L. ad Suesh, R. P. 99. A ew mehod o esmao o locao ad scale paamees. J. Sa. Pla. I. 30, 8-9. Tku, M.L. ad Vaugha, D.C A amly o sho-aled symmec dsbuos. Techcal Repo: McMase Uvesy, Caada. Tku, M.L., Wog, W.K., Vaugha, D.C. ad Ba, G.R Tme sees models oomal suaos: Symmec ovaos, J. Tme Sees Aalyss, Vaugha, D.C. 99b Epeced values, vaaces ad covaaces o ode sascs o Sude's dsbuo wh wo degees o eedom. Commu. Sa. Smul., Vaugha, D.C. 99. O he Tku-Suesh mehod o esmao. Commu. Sas.-Theoy Meh.,, Vaugha, D.C. 994 The eac values o he epeced values, vaaces ad covaaces o he ode sascs om he Cauchy dsbuo. J. Sa. Compu. Smul. 49, -3. 4