Hazard Rate Function Estimation Using Weibull Kernel

Transcription

1 Ope Joural of Statistics, 4, 4, Publised Olie September 4 i SciRes. ttp:// ttp://d.doi.org/.46/ojs Hazard Rate Fuctio Estimatio Usig Weibull Kerel Raid B. Sala, Hazem I. El Sek Amed, Iyad M. Aloubi Departmet of Matematics, Te Islamic Uiversity of Gaza, Gaza, Palestie Departmet of Matematics, Al Quds Ope Uiversity, Gaza, Palestie Academic Departmet, Uiversity College of Sciece ad Tecology, Gaza, Palestie rbsala@mail.iugaza.edu, saikamad@qou.edu, i.oubi@cst.ps Received July 4; revised 9 July 4; accepted August 4 Copyrigt 4 by autors ad Scietific Researc Publisig Ic. Tis work is licesed uder te Creative Commos Attributio Iteratioal Licese CC BY. ttp://creativecommos.org/liceses/by/4./ Abstract I tis paper, we defie te Weibull kerel ad use it to oparametric estimatio of te probability desity fuctio pdf ad te azard rate fuctio for idepedet ad idetically distributed iid data. Te bias, variace ad te optimal badwidt of te proposed estimator are ivestigated. Moreover, te asymptotic ormality of te proposed estimator is ivestigated. Te performace of te proposed estimator is tested usig simulatio study ad real data. Keywords Weibull Kerel, Hazard Rate Fuctio, Kerel Estimatio, Asymptotic Normality. Itroductio Hazard rate fuctios ca be used for several statistical aalyses i medicie, egieerig ad ecoomics. For istace, tey are commoly used we presetig results i cliical trials ivolvig survival data. Several metods for azard fuctio estimatio ave bee cosidered i te literature. Hazard fuctio estimatio by oparametric metods as a advatage i fleibility because o formal assumptios are made about te mecaism tat geerates te sample order or te radomess. Estimators of te azard fuctio based o kerel smootig ave bee studied etesively. For istace, see []-[5]. Te performace of te estimator at boudary poits differs from te iterior poits due to so-called boudary effects tat occur i oparametric curve estimatio problems; more specifically, te bias of te estimator at boudary poits. To remove tose boudary effects i kerel desity estimatio, a variety of metods ave bee developed i te literature. Some well-kow metods are summarized below: Te reflectio metod [6]-[8]. How to cite tis paper: Sala, R.B., El Sek Amed, H.I. ad Aloubi, I.M. 4 Hazard Rate Fuctio Estimatio Usig Weibull Kerel. Ope Joural of Statistics, 4, ttp://d.doi.org/.46/ojs.4.486

2 R. B. Sala et al. Te local liear metod [9] []. I [] tis problem is solved by replacig te symmetric kerels by asymmetric Gamma kerel wic ever assigs weigt outside te support. A lot of people wo care of estimatio do may specific distributios, suc as ormal, log-ormal, gamma ad iverse-gamma distributios, ad tis makes us pose a questio wic is: is tere ay oter distributios wic ca be used as a kerel i te estimatio ad te give us acceptable results? I tis paper, we propose te Weibull kerel wic also ever assigs weigt outside te support. Te Weibull distributio is a cotiuous probability distributio. It is amed after Waloddi Weibull, wo described it i detail i 95, altoug it was first idetified by Frcet 97 ad first applied by Rosi ad Rammler 9 to describe a particle size distributio. Tis paper is orgaised i five sectios. I te first sectio, we preset some iformatio about kerel smootig, azard fuctios ad te proposed kerel. I te secod sectio, we itroduce some defiitios ad relatios, ad state te coditios uder wic te results of te paper will be proved. I te tird sectio, we ivestigate te bias, variace ad optimal badwidt of te Weibull kerel estimator. I te fourt sectio, we ivestigate te asymptotic ormality of te Weibull kerel estimator of te pdf ad of te azard fuctio estimator. I te fift sectio, te performace of te proposed estimator will be tested via two applicatios, simulated ad real data.. Prelimiaries I tis sectio, we state te coditios uder wic te results of te paper will be proved. Also, we metio te defiitio of gamma fuctio ad some relatios related to it, te we will defie Weibull kerel. Coditios Let { i} i be a radom sample from a distributio of ukow probability desity fuctio f defied o [, suc tat f as a cotiuous secod derivative. 4 f d f < ad < d <. γ 4 f d ep l 8 > f 4γ d 4 is smootig parameter satisfyig + as. Te Gamma fuctio is defied to be te improper itegral:, were γ is Euler s costat. Γ t ep t d t, Te Taylor epasio about zero of Γ + ad Γ is available ad give by: Also, Γ γ. O te oter ad, otice tat: π Γ + γ+ γ + + o γ 6 π + γ + γ + o Γ 6 l Γ + Γ + Γ lim Γ + ep lim ep lim ep ep Γ + Γ γ 65

3 R. B. Sala et al. Moreover, te differece betwee γ ad ep γ γ by ep γ. I tis paper we cosider te followig Weibull kerel fuctio: gets small we, we ca approimate Γ + tγ + t Kw, t ep Γ +, t If a radom variable Y as a pdf K, t w, te E Y Γ + var Y Γ +, ad te variace is We propose te followig estimator of te probability desity fuctio f., te Weibull estimator, f ˆ K, X i. Bias, Variace ad Optimal Badwidt Propositio. Te bias of te proposed estimator is give by: Proof: were ˆ w i Γ + Bias f f + o Γ + ˆ w, d ξ E f K y f y y E f ξ follows a Weibull distributio wit scale parameter Γ + te epressio of te mea ad variace of Weibull distributio we deduce tat te mea is µ E ξ Γ + Var. Γ + Te Taylor epasio about f ξ is: ad V ξ So, ξ Hece, µ for f ξ f µ + f µ ξ µ + f µ ξ µ + o. + ξ + E fˆ f f var o. Γ + E f ξ f f o Γ +. ad sape parameter, ad from ˆ + +. Γ + f f + o Γ + ˆ Bias. 65

4 R. B. Sala et al. Propositio. Te variace of te proposed estimator is give by: Proof: were, ˆ Var f Γ + f + o. ˆ f K w t E Kw t E Kw t Var Var,,, Γ + tγ + tγ + Kw, t ep Usig te trasformatio Γ + tγ + tγ + t Γ + ep Γ + tγ + tγ + Γ + tγ + ep Γ + Γ + tγ + t Γ + Γ + t ep Γ + Γ + tγ + t Γ + t ep z t we get: Γ + z Γ + zγ + z Kw, z ep Γ + Terefore, were ad Γ + Γ + zγ + zγ + z ep for < z< E Kw, z E B, η f η, B, Γ + η follows as Weibull distributio wit mea µ E η V ad variace Γ + Var η Γ + 65

5 R. B. Sala et al. Te Taylor epasio of η f η is as follows: So, η f η µ f µ µ f µ µ f µ η µ Tis implies tat µ µ + µ µ + µ µ η µ + f f f o E η f f f f η + + Γ + + f + o Γ + Γ + + f Γ + Γ + f + f + f Γ + + o, Γ + E Bη f η Γ + f + Γ + f Γ + Terefore, Γ + + Γ + f + f + o Γ + ˆ Var f Γ + f f + o + o Γ + f + o. Optimal Badwidt: First of all, we will defie MSE ad Mea Itegrated Squared Error MISE as follows: Terefore, ˆ Γ + 8 MSE f f + o + Γ + f + o Γ + 4 Γ + Γ f + Γ + f + o Γ + 8 f + Γ + f 4 Γ

6 R. B. Sala et al. ˆ Γ + 8 f MISE f f d+ Γ + d 4 Γ + 4 Te Taylor epasio of 8 is give as follow: Terefore, we ca approimate MISE to be: 4 were, 4.4 l o + 8 l 8 l 8 ˆ I [ γ ] I ep γ MISE f + + l 8, f I f d ad I d. We will ow fid te optimal badwidt by miimizig.5 wit respect to, so we ave fˆ I γ d MISE ep l 8 γi+ γ I +..6 d 4 d MISE d Settig.6 equal zero yields te optimal badwidt fˆ γ I >..7 opt for te give pdf ad kerel: γ 4γI I ep l 8 opt.8 8γ I I additio,.7 proves tat tis value miimize.5. Substitutig.8 for i.5 gives te miimum MISE for te give pdf ad kere wic is give by: Note tat MISEopt l 8 4 4γ I 4 8γ I I I ep γ l 8 I ep γ 4γI I ep γ l opt deped o te sample size, te kerel ad o te ukow pdf. 4. Asymptotic Normality I tis sectio, we state te mai two teorems talkig about te asymptotic ormality for te proposed estimator ad a importat lemma wic we will use i te secod mai teorem. Defiitio. A azard rate fuctio is defied as te probability of a evet appeig i a sort time iterval. More precisely, it is defied as: S + > P X X r lim, >. Te azard rate fuctio ca be writte as te ratio betwee te pdf.. F. as follows: f r S Te kerel estimator of S. is Sˆ Fˆ were, F ˆ f ˆ u d u K u, X d u w i i Defiitio. We defied te proposed estimator for te azard rate fuctio to be:.9 f ad te survivor fuctio 655

7 R. B. Sala et al. rˆ fˆ Sˆ Teorem. Uder coditios,, ad, te followig olds Proof: Let V K, X i w i Γ + ˆ d f f f N, 4, i,,,. Te i. f ˆ V i, were Vi, i,,, are idepedet ad idetically distributios iid ad σ i <. We sow ow tat te Liapouov coditio is satisfied, tat is for some >, First of all, we ave: V + + V + EV E V lim + + Γ + + σ + + Γ + tγ + tγ + ep +, t + + Γ + + zγ + zγ + + ep, z + zγ + Γ + zγ + zγ + ep z Γ + z Γ + Γ + Γ + + z ep z z ep α z + Γ + Γ + Γ + were, + + Γ + α + +. Te we ave: t were, + + y y Γ + Γ + Γ + E V α y ep f y dy + E αξ f ξ, ξ follows a Weibull distributio wit mea µ E ξ ad variace 656

8 R. B. Sala et al. Te Taylor epasio of ξ f ξ ξ Γ + Γ + var. + about te mea µ is as follows: ξ f ξ f f f ξ f + + f + f + o + + ξ E ξ f ξ f f were H +. Terefore, Γ + + f + f + o Γ f + H + f Γ +, Γ + + H f + f + o Γ + + E V f o. Now substitutig, te followig is old: Hece, V Γ + f + o Var EV + Γ σ σ EV E V f V V Γ + f Γ + f + Γ + f Te last term vaises as, sice coditio 4 implies tat ad. Also, te remaiig compoet of te last term are bouded from coditio. Lemma. Uder coditios, ad te followig olds F ˆ p F. 657

9 R. B. Sala et al. Proof: First of all, we ave from te defiitio of ˆF te followig relatios: were Terefore, ˆ EF K, d d, dd d wu y uf y y K wu y f y y u E f ξ u Γ + f u + u f u du+ o F + o Γ + ξ is defied as i Propositio. Tis implies tat, EF ˆ F o ˆ p EF F o O te oter ad, ˆF ca be writte as follows: W K u, y d. u Now, give ε >, >, te we ave: were i w F ˆ K u, y d u W i w i i + E Fˆ + EFˆ + ε + + ˆ ˆ P F EF ε E Fˆ EFˆ > ε ε i + E Wi EWi E Wi ε EWi + + ε + i i Te secod term vaises as ad, sice from. we ave + EWi o + + i,. Furter, by replacig wit i. ad assumig tat we ave: τ, + Γ + P F EF E Wi i + + ˆ ˆ + > ε ε + ε w + ε w τ, + + k u, y duf y dy k u, y f y dd yu ε f u + o d F EF ˆ ˆ p

10 R. B. Sala et al. ˆ ˆ ˆ + ˆ, te by. ad.4, we ave: Sice F F F EF EF F F F F EF EF F ˆ ˆ ˆ ˆ p + Tis complete te proof of te lemma. Teorem. Uder coditios,, ad, te followig olds d rˆ r N, 4 Γ + r S Proof: fˆ f rˆ r Sˆ S Γ + Γ + Γ + Γ + fˆ f f f + Sˆ Sˆ S Sˆ Sˆ S ˆ f f f + f Sˆ f ˆ.5 fˆ f Sˆ S ˆ S S S Γ + Γ + d r N, 4 S Note tat te secod term vaises by., ad te first term is asymptotically ormally distributed by.. Moreover, from. ad.5 too we ave: ad, Terefore, Γ + f + f ˆ E f Γ + E rˆ + o E Sˆ S f r + + o rˆ Γ + Γ + S Γ + Γ + S f Bias + o Var rˆ Γ + r + o S 659

11 R. B. Sala et al. 5. Applicatios I tis sectio, te performace of te proposed estimator i estimatig te pdf ad azard rate fuctio is tested upo two applicatios usig a simulated ad real life data. 5.. A Simulatio Study A sample of size from te epoetial distributio wit pdf f ep te badwidt usig te relatio opt 5 is simulated. We computed.79 R,.6 see [8] page 47 ad it equals Te desity ad te azard fuctios were estimated usig te Weibull estimator. Te estimated values ad te true epoetial pdf are plotted i Figure a, tis figure sows tat te performace of te Weibull estimator is acceptable at te boudary ear te zero. I te iterior te beavior of te pdf estimator becomes more similar as we get away from zero. Also Figure b sows tat te performace of te Weibull estimator of te azard fuctio is acceptable at te boudary ear te zero wic we cocer o. Te mea squared error MSE of proposed estimator of te desity fuctio is equal to.976 ad for te azard fuctio is evaluated for te iterval [,.5] because we cocer about closest values to zero ad is equal to Real Data I tis subsectio, we used te suicide data give i Silverma [8], to eibit te practical performace of te Weibull estimator. Te data gives te legts of te treatmet spells i days of cotrol patiets i suicide study. We used te logaritm of te data to draw Figure a ad Figure b usig badwidt equals.484 wic computed by.6, tese figures eibit te two estimated fuctios of te probability desity ad azard rate fuctios, respectively. 6. Commet ad Coclusio I tis paper, we ave proposed a ew kerel estimator of te azard rate fuctio for iid data based o te Weibull kerel wit oegative support; we sowed tat te bias depeds o te smootig parameter ad te estimated poit, ad it goes to zero as, also it gets small for te values of closed to zero. Te variace was ivestigated ad we oticed tat it depeds also o ad. O te oter ad, it goes to zero as, ad gets large at te values of close to zero. Moreover, te optimal badwidt ad te asymptotic ormality were ivestigated. a Figure. a Te Weibull kerel estimator of te desity fuctio; b Te azard rate fuctio for te simulated data of te epoetial distributio. b 66

12 R. B. Sala et al. a Figure. a Te Weibull kerel estimator of te desity fuctio; b Te azard rate fuctio for te suicide data. I additio, te performace of te proposed estimator is tested i two applicatios. I a simulatio study usig epoetial sample we oticed tat te performace of te proposed estimator is acceptable, ad gives a small MSE. Usig real data, we eibited te practical performace of te Weibull estimator. Refereces [] Sala, R. Hazard Rate Fuctio Estimatio Usig Iverse Gaussia Kerel. Te Islamic Uiversity of Gaza Joural of Natural ad Egieerig Studies,, [] Sala, R. Estimatig te Desity ad Hazard Rate Fuctios Usig te Reciprocal Iverse Gaussia Kerel. 5t Applied Stocastic Models ad Data Aalysis Iteratioal Coferece, Mataro Barceloa, 5-8 Jue. [] Scaillet, O. 4 Desity Estimatio Usig Iverse ad Reciprocal Iverse Gaussia Kerels. Noparametric Statistics, 6, 7-6. ttp://d.doi.org/.8/ [4] Watso, G. ad Leadbetter, M. 964 Hazard Aalysis I. Biometrika, 5, ttp://d.doi.org/.9/biomet/ [5] Rice, J. ad Roseblatt, M. 976 Estimatio of te Log Survivor Fuctio ad Hazard Fuctio. Sakya Series A, 8, [6] Clie, D.B.H. ad Hart, J.D. 99 Kerel Estimatio of Desities of Discotiuous Derivatives. Statistics,, ttp://d.doi.org/.8/ [7] Scuster, E.F. 985 Icorporatig Support Costraits ito Noparametric Estimators of Desities. Commuicatios i Statistics, Part A Teory ad Metods, 4, -6. ttp://d.doi.org/.8/ [8] Silverma, B.W. 986 Desity Estimatio for Statistics ad Data Aalysis. Capma ad Hall, Lodo. ttp://d.doi.org/.7/ [9] Ceg, M.Y., Fa, J. ad Marro, J.S. 997 O Automatic Boudary Correctios. Te Aals of Statistics, 5, ttp://d.doi.org/.4/aos/59477 [] Zag, S. ad Karuamui, R.J. 998 O Kerel Desity Estimatio ear Edpoits. Joural of Statistical Plaig ad Iferece, 7, -6. ttp://d.doi.org/.6/s [] Ce, S.X. Probability Desity Fuctio Estimatio Usig Gamma Kerels. Aals of te Istitute of Statistical Matematics, 5, ttp://d.doi.org/./a: b 66

13