Study of the Left Censored Data from the Gumbel Type II Distribution under a Bayesian Approach

Transcription

1 Joual of Mode Appled Stattcal Method Volume 5 Iue Atcle Study of the Left Ceoed Data fom the Gumbel Type II Dtbuto ude a Bayea Appoach Tabaum Naz Sdhu Quad--Azam Uvety, Ilamabad, Pata, dhuqau@gmal.com Navd Feoze Rpha Iteatoal Uvety, Ilamabad, Pata, avdfeoz@gmal.com Muhammad Alam Rpha Iteatoal Uvety, Ilamabad, Pata, alamdqu@yahoo.com Follow th ad addtoal wo at: Pat of the Appled Stattc Commo, Socal ad Behavoal Scece Commo, ad the Stattcal Theoy Commo Recommeded Ctato Sdhu, Tabaum Naz; Feoze, Navd; ad Alam, Muhammad (6) "Study of the Left Ceoed Data fom the Gumbel Type II Dtbuto ude a Bayea Appoach," Joual of Mode Appled Stattcal Method: Vol. 5 : I., Atcle 0. DOI: 0.37/jmam/470 Avalable at: Th Regula Atcle bought to you fo fee ad ope acce by the Ope Acce Joual at DgtalCommo@WayeState. It ha bee accepted fo cluo Joual of Mode Appled Stattcal Method by a authozed edto of DgtalCommo@WayeState.

2 Joual of Mode Appled Stattcal Method Novembe 6, Vol. 5, No., -34. do: 0.37/jmam/470 Copyght 6 JMASM, Ic. ISSN Study of the Left Ceoed Data fom the Gumbel Type II Dtbuto ude a Bayea Appoach Tabaum Naz Sdhu Quad--Azam Uvety Ilamabad, Pata Navd Feoze Rpha Iteatoal Uvety Ilamabad, Pata Muhammad Alam Rpha Iteatoal Uvety Ilamabad, Pata Baed o left type II ceoed ample fom a Gumbel type II dtbuto, the Baye etmato ad coepodg of the uow paamete wee obtaed ude dffeet aymmetc lo fucto, aumg dffeet fomatve ad o-fomatve po. Elctato of hype-paamete though po pedctve appoach ha alo bee dcued. The epeo fo the cedble teval ad poteo pedctve dtbuto have bee deved. Compao of thee etmato ae made though mulato tudy ug umecal ad gaphcal method. Keywod: dtbuto Left ceog, lo fucto, cedble teval, poteo pedctve Itoducto Gumbel type II dtbuto vey ueful lfe tetg. Kotz ad Nadaajah (00) have gve a bef chaactezato of the Gumbel type II dtbuto. Co, G, Geco, ad Veazza (0) tuded the mamum lelhood (ML) algothm ad Came-Rao (CR) boud fo the locato ad cale paamete of the Gumbel dtbuto. Moua, Jahee, ad Ahmad (0) codeed the Bayea etmato to aalyze both paamete of the Gumbel dtbuto baed o ecod value. The pobablty dety fucto of the Gumbel dtbuto of the ecod d gve by f ep, 0,, 0. () T. N. Sdhu the Depatmet of Stattc. Emal at: dhuqau@gmal.com. N. Feoze the Depatmet of Stattc. Emal at: avdfeoz@hotmal.com. M. Alam the Depatmet of Stattc. Emal at: alamdqu@yahoo.com.

3 SINDHU ET AL. The coepodg cumulatve dtbuto fucto : ep, 0,, 0. F () The paamete υ (beg ow) a hape paamete of the model, ad τ the cale paamete. The ue of a Bayea appoach allow both ample ad po fomato to be copoated to the tattcal aaly, whch wll mpove the qualty of the feece ad pemt a educto ample ze. The deco-theoetc vewpot tae to accout addtoal fomato coceg the poble coequece of deco (quatfed by a lo fucto). The am of th to code the tattcal aaly of the uow paamete whe the data ae left ceoed fom the Gumbel dtbuto of the ecod d. Thee a wdepead applcato ad ue of left-ceog o left-ceoed data uvval aaly ad elablty theoy. Fo eample, medcal tude patet ae ubject to egula eamato. Dcovey of a codto oly tell u that the oet of ce fell the peod ce the pevou eamato ad othg about the eact date of the attac. Thu the tme elaped ce oet ha bee left ceoed. Smlaly, code left-ceoed data whe etmatg fucto of eact polcy duato wthout owg the eact date of polcy ety; o whe etmatg fucto of eact age wthout owg the eact date of bth. Cobu, McBde ad Zlle (0) faced th poblem due to the cdece of a hghe popoto of ual chlde whoe pell wee left ceoed (.e., thoe chlde who eteed the ample uued), ad who emaed uued thoughout the ample. A aothe eample, job duato mght be complete becaue the begg of the job pell ot obeved, whch a cdece of left ceog (Bagge, 05). Lelhood Fucto ad Poteo Dtbuto Let X ( + ),, X () be the lat - ode tattc fom a adom ample of ze followg Gumbel type II dtbuto. The the jot pobablty dety fucto of X ( + ),, X () gve by! f,..., ;, F f... f! 0 ep, (3) 3

4 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION whee =, ad ep. Po ad Poteo Dtbuto The ufom po aumed to be p, 0. (4) The poteo dtbuto ude the ufom po fo the left ceoed data : p 0 ep 0, 0. (5) The fomatve po fo the paamete τ aumed to be epoetal dtbuto: w, 0, 0. p we w (6) The poteo dtbuto ude the aumpto of epoetal po : p 0 ep w 0, 0 w (7) The fomatve po fo the paamete τ aumed to be gamma dtbuto: 4

5 SINDHU ET AL. b,,, 0. a a b p e a b a (8) The poteo dtbuto ude the aumpto of gamma po fo the left ceoed data : p 0 a ep b 0, 0. a a b (9) The fomatve po fo the paamete τ aumed to be vee Levy dtbuto: c c p e, c, 0. (0) The poteo dtbuto ude the vee Levy po fo the left ceoed data : p 0 c, 0 c ep 0 () Baye Etmato ad Poteo R ude Dffeet Lo Fucto Code the devato of the Baye etmato ad coepodg poteo ude dffeet lo fucto. The Baye etmato ae evaluated ude pecautoay lo fucto (PLF), weghted quaed eo lo fucto (WSELF), quaed-log eo lo fucto (SLELF), ad etopy lo fucto (ELF). The 5

6 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION Baye etmato ad coepodg poteo ude dffeet lo fucto ae gve the Table. Table. Baye etmato ad poteo ude dffeet lo fucto Lo Fucto =, ˆ ˆ PLF: L Baye Etmato Poteo R ˆ ˆ WSELF: ˆ SLELF: l ˆ l ep El E E E E E E l l E E ELF: E E l ˆ ˆ l E l The Baye etmato ad poteo ude ufom po ae: ˆ PLF, ˆPLF 0 0 6

7 SINDHU ET AL ˆ, WSELF ˆ. WSELF 0 0 ep ˆ, SLELF 0 0 ˆ. SLELF 0 0 ˆ, ELF

8 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION 0 0 ˆ ELF l. 0 0 The Baye etmato ad poteo ude the et of po ca be obtaed a mla mae. Baye Cedble Iteval fo the Left Ceoed Data The Bayea cedble teval fo type II left ceoed data ude fomatve ad o-fomatve po, a dcued by Saleem ad Alam (09) ae peeted the followg. The cedble teval fo type II left ceoed data ude all po ae: 0 0 Ufom w w Epoetal 0 0 w w a a 0 0 b b Gamma a a 0 0 b b 8

9 SINDHU ET AL c c ILevy 0 0 c c Elctato Code a pobablty elctato method ow a po pedctve elctato. Pedctve elctato a method fo etmatg hype-paamete of po dtbuto by vetg coepodg po pedctve dtbuto. Elctato of hype-paamete fom the po p(τ) coceptually dffcult ta becaue we ft have to detfy po dtbuto ad the t hype-paamete. The po pedctve dtbuto ued fo the elctato of the hype-paamete whch compaed wth the epet' judgmet about th dtbuto ad the the hypepaamete ae choe uch a way o a to mae the judgmet agee cloely a poble wth the gve dtbuto (ee Gmhaw, 993; Kadae, 9; O'Haga et al., 06; Gmhaw, Collg, Lae, & Hut, 0; Jeo, 05; ad Leó, Vázquez-Polo, & Gozález, 03). Accodg to Alam (03), the method of aemet to compae the pedctve dtbuto wth epet' aemet about th dtbuto ad the to chooe the hype-paamete that mae the aemet agee cloely wth the membe of the famly. He dcue thee mpotat method to elct the hypepaamete: () va the po pedctve pobablte () va elctato of the cofdece level () va the pedctve mode ad cofdece level. We wll ue the po pedctve appoach by Alam (03). Po pedctve dtbuto The po pedctve dtbuto : The pedctve dtbuto ude epoetal po : p y p y p d () 0 9

10 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION Afte ome mplfcato t educe a 0 p y w y y w d (3) ep w p y y w y, y 0. (4) The pedctve dtbuto ude gamma po : a ab, 0 y. a p y y b y c, 0 y. 3 y c y 3 p y (5) (6) By ug the method of elctato defed by Alam (03), we obta the followg hype-paamete w = , a = 0.59, b = ad c = Poteo Pedctve Dtbuto The pedctve dtbuto cota the fomato about the depedet futue adom obevato gve pecedg obevato. The eade dee moe detal ca ee Baal (07). The poteo pedctve dtbuto of the futue obevato y = + p y p p y d (7) 0

11 SINDHU ET AL. Whee p y ep, the futue obevato dety ad p (τ ) the poteo dtbuto obtaed by copoatg the lelhood wth the epectve po dtbuto. The poteo pedctve dtbuto of the futue obevato y = + ude ufom po p y 0 0 y y, y 0. (8) The poteo pedctve dtbuto of the futue obevato y = + ude epoetal po p y 0 0 y w y w, y 0. (9) The poteo pedctve dtbuto of the futue obevato y = + ude gamma po p y 0 a 0 a y b y a b, y 0. () The poteo pedctve dtbuto of the futue obevato y = + ude Ivee-Levy po

12 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION p y 0 Smulato Study 0 3 y c y c, y 0. () Smulato ca be helpful ad a llumatg way to appoach poblem Bayea aaly. Bayea poblem of updatg etmate ca be hadled ealy ad taght fowadly wth mulato. Becaue the dtbuto fucto of the Gumbel type II dtbuto ca be epeed, a well a t vee cloed fom, the veo method of mulato taghtfowad to mplemet. The tudy wa caed out fo dffeet value of (, ) ug τ.5 ad υ = 0.5. Ceog ate ae aumed to be 5% ad 0%. Sample ze vaed to obeve the effect of mall ad lage ample o the etmato. Chage the etmato ad the have bee detemed whe chagg the lo fucto ad the po dtbuto of τ whle eepg the ample ze fed. All thee eult ae baed o 5,000 epetto. Table -6 gve the etmated value of the paamete, poteo ad 95% cofdece lmt (Lowe Cofdece Lmt (LCL) ad Uppe Cofdece Lmt (UCL)) fo the paamete. The eult ae ummazed the followg Table ad Fgue -8. The amout of poteo have bee peeted the paethe the table.

13 SINDHU ET AL. Table. Baye etmate ad the poteo ude PLF fo τ Ufom Po No Ceog 5% Ceog 0% Ceog.7379 (0.5898).6779 ( ).645 ( ) (0.0350).5638 (0.0573).54 (0.8643).598 (0.09).54 (0.07).57 ( ) ( ) ( ).8853 ( ).0086 ( ).37 (0.06).848 (0.0653).3737 ( ).78 ( ).933 ( ).0477 (0.059).3 (0.095) ( ) (0.0779) ( ) (0.0383) ( ) Epoetal Po Gamma Po.96 (0.386) (0.07) (0.049) ( ).9773 ( ) (0.0755) (0.050).6658 ( ).450 ( ).504 ( ) Ivee Levy Po ( ).9963 ( ).666 (0.033).3593 (0.0934).466 (0.0443) (0.870) ( ) (0.0447) (0.0458) (0.035) ( ) ( ) ( ) (0.0434) ( ) ( ).85 (0.0554) (0.0435).675 ( ).790 (0.0889).493 (0.0794) ( ) ( ).5509 ( ).687 (0.0755) 3

14 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION Table 3. Baye etmate ad the poteo ude WSELF fo τ Ufom Po No Ceog 5% Ceog 0% Ceog.669 (0.335) ( ).553 ( ) ( ).5670 (0.0567) (0.8978).48 (0.070) (0.048) ( ) ( ) ( ) ( ).9554 ( ).0765 (0.063).446 (0.0630).4650 ( ).6650 (0.0455).8683 (0.030) (0.053).89 (0.093) 3.08 ( ) ( ) (0.0590) ( ) ( ) Epoetal Po Gamma Po.934 ( ) ( ).8557 ( ).788 ( ).643 ( ) ( ).9 ( ).5507 ( ).36 ( ) (0.0550) Ivee Levy Po.37 (0.0690).7489 ( ).0 ( ).3347 ( ).49 (0.0470) (0.3) (0.0909) (0.068) ( ) ( ) (0.5847) ( ) 3.70 ( ) 3.34 ( ) ( ).5586 (0.0755).59 (0.055).4499 ( ).6334 (0.0344) ( ).3867 (0.0787) ( ).367 (0.076).5099 (0.036).68 ( ) 4

15 SINDHU ET AL. Table 4. Baye etmate ad the poteo ude SLELF fo τ Ufom Po No Ceog 5% Ceog 0% Ceog ( ).8 ( ).5487 (0.0659) (0.04) ( ).46 ( ).4664 ( ).4773 (0.0659).488 (0.04).4975 ( ).3708 ( ).789 (0.058).9830 ( ).086 (0.0554).664 ( ).7054 ( ).6935 ( ).9054 ( ).047 (0.0499).67 ( ) Epoetal Po (0.057) ( ) (0.0739) (0.0307) 3.09 (0.0047) (0.057) ( ) (0.0739) (0.0307).85 (0.0047) Gamma Po ( ).9883 ( ).3 (0.0765) (0.038) ( ) Ivee Levy Po.3443 (0.0569) (0.0634).974 ( ).9894 (0.0358).058 (0.0056) (0.054) ( ) ( ) ( ) (0.0050) 3.36 (0.054) 3.38 ( ) 3.04 ( ) ( ) (0.0050) ( ).5504 (0.047).4558 ( ) ( ).770 (0.054).486 ( ).036 ( ).343 (0.0858).56 ( ) (0.0) 5

16 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION Table 5. Baye etmate ad the poteo ude ELF fo τ Ufom Po No Ceog 5% Ceog 0% Ceog (0.0479) (0.0448) ( ).587 ( ).54 ( ).5650 (0.0479).5434 (0.0448) ( ).4848 ( ) ( ).3397 ( ).766 (0.0763).9457 ( ).0737 ( ).5873 ( ).3549 (0.054) (0.05) (0.0083).04 ( ) ( ) Epoetal Po Gamma Po Ivee Levy Po ( ) (0.0508) ( ) ( ) ( ) ( ).5858 (0.0508).5638 ( ).555 ( ) ( ).4488 ( ) (0.0456).469 (0.0083) ( ).4750 ( ).3738 (0.0459) (0.034).00 ( ).6947 (0.0064).3956 ( ) (0.065) (0.0576) ( ) (0.0067) ( ) (0.065) (0.0576) ( ) (0.0067).9496 ( ).596 ( ).58 (0.0955).4467 (0.0047).6396 (0.007) (0.0048).3907 (0.0389) ( ).300 ( ) (0.006).6530 ( ) 6

17 SINDHU ET AL. Table 6. The 95% cedble teval fo τ.5. Ufom Po Lowe Lmt Uppe Lmt Dffeece Epoetal Po Gamma Po Ivee Levy Po Gaphcal Repeetato of Poteo R ude Dffeet Po The gaph eveal that poteo ude dffeet fomatve ad o fomatve po. It obeved that both the po (ufom ad epoetal) yeld the appomately the detcal poteo feece ude ELF ad SLELF. 7

18 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION Fgue. Effect of poteo ude PLF wth o ceog Fgue. Effect of poteo ude PLF wth 0% ceog 8

19 SINDHU ET AL. Fgue 3. Effect of poteo ude WSELF wth o ceog Fgue 4. Effect of poteo ude WSELF wth 0% ceog 9

20 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION Fgue 5. Effect of poteo ude SLELF wth o ceog Fgue 6. Effect of poteo ude SLELF wth 0% ceog 30

21 SINDHU ET AL. Fgue 7. Effect of poteo ude ELF wth o ceog Fgue 8. Effect of poteo ude ELF wth 0% ceog 3

22 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION Cocluo The mulato tudy dplayed ome teetg popete of the Baye etmate. The ude ad lo fucto ae educed a the ample ze ceae. The effect of ceog o etmato of τ the fom of oveetmato ude ufom ad epoetal po ad udeetmato aumg gamma ad vee Levy po. Lage degee of ceog eult bgge ze of ove o udeetmato. Howeve, the paamete τ ethe udeetmated o oveetmated depedg upo the po dtbuto to be ued whe ceog ot doe. The etet of th ove o ude etmato dectly popotoal to amout of ceog ate ad veely popotoal to the ample ze. Futhe, the ceae ample ze educe the poteo of τ. Aothe teetg ema coceg the of the etmate that ceag (deceag) the ceog ate ceag (educe) the of the etmate ude ad lo fucto. The pefomace of quaed-log eo lo fucto ad etopy lo fucto depedet of choce of paametc value. I compao of fomatve po ad the ufom po, the vee Levy po povde the bette etmate a the coepodg ae leat ude ad lo fucto ecept ELF ad SLELF. Although the ufom ad the epoetal po ae equally effcet ude ELF ad SLELF, theefoe they poduce moe effcet etmate a compaed to the othe fomatve po. The cedble teval ae accodace wth the pot etmate, that, the wdth of cedble teval veely popotoal to ample ze. Fom the Table 6, appeded above, t ca be evealed that the effect of the po fomato the fom of aowe wdth of teval. The cedble teval aumg vee Levy po much aowe tha the cedble teval aumg fomatve ad o-fomatve po. It the ue of po fomato that mae a dffeece tem of ga peco. To ee the effect of the poteo aumg dffeet po Fgue -8 ae pepaed. It obeved fom all the fgue that poteo deceae wth the ceae ample ze ude all lo fucto. It evdet fom Fgue 5-8 that behavo of poteo mla all apect. The tudy ca futhe be eteded by codeg geealzed veo of the dtbuto ude vaety of ccumtace. 3

23 SINDHU ET AL. Refeece Alam, M. (03). A applcato of po pedctve dtbuto to elct the po dety. Joual of Stattcal Theoy ad Applcato, (), Bagge, J. (05). Wage gowth ad tuove Dema. Dema: Uvety of Aahu. Baal, A. K. (07). Bayea Paametc Ifeece. New Delh, Ida: Naoa Publhg Houe. Cobu, A. F., McBde, T. D., & Zlle, E. C. (0). Patte of Health Iuace Coveage amog Rual ad Uba Chlde. Medcal Cae Reeach ad Revew, 59(3), 7-9. do: 0.77/ Co, G., G, F., Geco, M. V., & Veazza, L. (0). Came-Rao boud ad etmato of the paamete of the Gumbel dtbuto. IEEE Taacto o Aeopace ad Electoc Sytem, 3(3), -4. do: 0.09/ Gmhaw, S. D. (993). Computg mamum lelhood etmate fo the geealzed Paeto dtbuto. Techometc, 35(), do: 0.0/ Gmhaw, S. D., Collg, B. J., Lae, W. A., & Hut, C. R. (0). Elctg Facto Impotace a Deged Epemet. Techometc, 43(), do: 0.98/ Jeo, D. (05). The elctato of pobablte: A evew of the tattcal lteatue. BEEP Wog Pape. Depatmet of Pobablty ad Stattc, Sheffeld: Uvety of Sheffeld. Kadae, J. B. (9). Pedctve ad Stuctual Method fo Elctg Po Dtbuto. I H. Jeffey & A. Zelle (Ed.), Bayea aaly ecoometc ad tattc: Eay hoo of Haold Jeffey. Amtedam: Noth-Hollad. Kotz, S., & Nadaajah, S. (00). Eteme value dtbuto. Theoy ad applcato. Lodo: Impeal College Pe. Leó, C. J., Vázquez-Polo, F. J., & Gozález, R. L. (03). Elctato of Epet Opo Beeft Tafe of Evometal Good. Evometal ad Reouce Ecoomc, 6(), 99-. do: 0.03/A: Moua, M. A M. A., Jahee, Z. F., & Ahmad, A. A. (0). Bayea etmato, pedcto ad chaactezato fo the Gumbel model baed o 33

24 LEFT CENSORED DATA FROM THE GUMBEL TYPE II DISTRIBUTION ecod. Stattc: A Joual of Theoetcal ad Appled Stattc, 36(), do: 0.0/ O'Haga, A., Buc, C. E., Daehhah, A., Ee, J. R., Gathwate, P. H., Jeo, D. J., Raow, T. (06). Uceta Judgemet: Elctg epet pobablte. Chchete: Joh Wley & So. Saleem, M. & Alam, M. (09). O Bayea aaly of the Raylegh uvval tme aumg the adom ceo tme. Pata Joual of Stattc, 5(),