Bayesian and Frequentist Prediction Using Progressive Type-II Censored with Binomial Removals

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1 Intellgent Infoaton Manageent Publhed Onlne eptebe 3 ( Bayean Fequentt Pedcton Ung Pogeve Type-II Cenoed w Bnoal Reoval Ahed A olan Ahed H Abd Ellah Nae A Abou-Elheggag Rahad M El-aghee 3* Faculty of cence Ilac Unvety Madnah aud Aaba Maeatc Depatent Faculty of cence ohag Unvety ohag Egypt 3 Maeatc Depatent Faculty of cence Al-Azha Unvety Cao Egypt Eal: * Rahada@ahooco Receved June 3 3; eved July 7 3; accepted Augut 9 3 Copyght 3 Ahed A olan et al Th an open acce atcle dtbuted unde e Ceatve Coon Attbuton Lcene whch pet unetcted ue dtbuton epoducton n any edu povded e ognal wo popely cted ABTRACT In atcle we tudy e poble of pedctng futue ecod ode tattc (two-aple pedcton) baed on pogeve type-ii cenoed w o eoval e nube of unt eoved at each falue te ha a dcete bnoal dtbuton We ue e Baye pocedue to deve bo pont nteval bound pedcton Bayean pont pedcton unde yetc yetc lo functon dcued The axu lelhood (ML) pedcton nteval ung plug-n pocedue fo futue ecod ode tattc ae deved An exaple dcued to llutate e applcaton of e eult unde cenong chee Keywod: Bayean Pedcton; Bu-X Model; Pogeve Cenong; Ro Reoval Intoducton In any pactcal poble of tattc one whe to ue e eult of pevou data (pat aple) to pedct a futue obevaton (a futue aple) fo e ae populaton One way to do to contuct an nteval whch wll contan e futue obevaton w a pecfed pobablty Th nteval called a pedcton nteval Pedcton ha been appled n edcne engneeng bune oe aea a well Hahn Meee [] have ecently dcued e uefulne of contuctng pedcton nteval Bayean pedcton bound fo a futue obevaton baed on cetan dtbuton have been dcued by eveal auo Bayean pedcton bound fo futue obevaton fo e exponental dtbuton ae condeed by Dunoe [] Lngappaah [3] Evan Ng [4] Al-Huan Jaheen [5] Bayean pedcton bound fo futue lfete unde e Webull odel have been deved by Evan Ng [67] Bayean pedcton bound fo obevable havng e Bu type-xii dtbuton wee obtaned by Ng [8] Al-Huan Jaheen [9] Al Moua Jaheen [] Pedcton wa evewed by Patel [3] Nagaja [4] Kany Nelon [5] Al- Huan [6] fo detal on e htoy of tattcal * Coepondng auo pedcton analy applcaton ee fo exaple Atchon Dunoe [7] Gee [8] Bayean pedcton bound fo e Bu type-x odel baed on ecod have been deved fo Al Moua [9] Bayean pedcton bound fo e caled Bu type X odel wee obtaned by Jaheen AL-Mataf [] Bayean pedcton w Outle o aple ze fo e Bu-X odel wa obtaned by olan [] Bayean pedcton bound fo ode tattc n e one two-aple cae fo e Bu type X odel wee obtaned by ataw Abu-alh [] Recently Ahad Balahnan [3] dcued how one can pedct futue uual ecod (ode tattc) fo an ndependent -equence baed on ode tattc (uual ecod) fo an ndependent X-equence developed nonpaaetc pedcton nteval Ahad M Motafaee [4] Ahad et al [5] obtaned pedcton nteval fo ode tattc a well a fo e ean lfete fo a futue aple baed on obeved uual ecod fo an exponental dtbuton ung e clacal Bayean appoache epectvely The et of e pape a follow In ecton we peent oe pelnae a e odel po e poteo dtbuton In ecton 3 Bayean pedctve dtbuton fo e futue lowe ecod (two-aple pedcton) baed on pogeve type-ii cenoed w Copyght 3 cre

2 A A OLIMAN ET AL 63 o eoval In ecton 4 e ML pedcton bo pont nteval pedcton ung plug-n pocedue ae deved In ecton 5 Bayean pedctve dtbuton fo e futue ode tattc baed on pogeve type-ii cenoed w o eoval In ecton 6 e ML pedcton bo pont nteval pedcton ung plugn pocedue fo e futue ode tattc ae deved A pactcal exaple ung geneatng data et Pogevely type-ii cenoed o aple fo Bu-X dtbuton a ulaton tudy ha been caed out n ode to copae e pefoance of dffeent eod of pedcton ae peented n ecton 8 Fnally we conclude e pape n ecton 8 The Model Po Poteo Dtbuton Let o vaable X have an Bu-X dtbuton w Paaete e pobablty denty functon e cuulatve dtbuton functon of X ae epectvely ( ) exp f x xexp x exp x x F x x x uppoe at x R x R x R denote a pogevely type-ii cenoed aple x x x w pe-detened nube of eoval ay R R R e condtonal lelhood functon can be wtten a ; L x R C f x F x n fo () () C n n n n 3 ubttutng () () nto (3) we get Gexp ln U (3) G U expx (4) If e paaete unnown fo e In-lelhood functon gven by (3) e MLE MLE can be obtaned by e followng equaton G ln ln ln U (5) The ft devatve of ln w epect to ln G ln U (6) ln ettng we get e axu lelhood etato of a e followng MLE G lnu conde a gaa conjugate po fo n e fo (7) π exp (8) Fo (3) (8) e condtonal poteo (pdf) of gven by π x q G exp lnu q 3 Bayean Pedcton fo Recod Value uppoe n ndependent te ae put on a tet e lfete dtbuton of each te gven by () Let X X X3 X be e odeed -falue obeved unde e type-ii pogevely cenong plan w bnoal eoval R R at be a econd ndependent aple (of ze ) of futue lowe ecod obeved fo e ae dtbuton (futue aple) Ou a to ae Bayean pedcton about e en e agnal pdf of V gven by ee Ahad M Motafaee [4] F! log F fv f exp exp log U exp exp f exp exp U q Applyng () () n () we obtan q V (9) () () () f exp q (3) Copyght 3 cre

3 64 A A OLIMAN ET AL Cobnng e poteo denty (9) w (3) ntegatng out we obtan e Baye pedctve denty x V x f f B d q G q q (4) The Bayean pedcton bound fo e futue ae obtaned by evaluatng P t x fo oe gven value of t It follow fo (4) at P t x f V x d t I B t q t t q q (5) d (6) The pedctve bound of a two-ded nteval w cove % fo e futue lowe ecod ay u obtaned by olvng e followng two equaton fo e lowe L uppe U bound: B B L U I G q (7) (8) L IU ae gven by Equaton (6) Now by ung (4) e Bayean pont pedcton of e futue lowe ecod value unde E (B) LINEX lo functon (BL) ae gven epectvely a x f V d B B (9) x Log exp c BL f V d c Log c B q q q () I d () q exp c q q I d () One can ue a nuecal ntegaton technque to get e above ntegaton gven by () () pecal cae: In pecal cae t potant to pedct e ft unobeved lowe ecod value When = n (7) (8) e lowe uppe Bayean pedcton bound w cove of ae obtaned fo e nuecal oluton of e followng equaton B L B U I (3) (4) L IU gven by Equaton (6) olvng e eultng equaton nuecally 4 ML Pedcton fo Recod Value The coonly ued fequentt appoache uch a e axu lelhood etate e plug-n pocedue whch to ubttute a pont etate of e unnown paaete nto e pedctve dtbuton ae evewed dcued In ecton e ML pedcton bo pont nteval ung plug-n pocedue fo futue lowe ecod baed on pogeve type-ii cenoed aple defned by () By eplacng n e agnal pdf of V (3) by whch we can fnd t fo e nuecal oluton of e Equaton (7) en Copyght 3 cre

4 A A OLIMAN ET AL 65 f q V exp q (5) The ML pedcton bound fo e futue ae obtaned by evaluatng P t x fo oe gven value of t It follow fo (5) at P t x f d Ingaa ; t V t Ingaa ; t t functon defned by Ingaa ; t t (6) e ncoplete gaa t z exp tz d z t log exp t (7) The pedctve bound of a two-ded nteval w cove fo e futue lowe ecod ay u obtaned by olvng e followng two equaton fo e lowe L uppe U bound: Ingaa L ; (8) Ingaa ; U (9) pecal cae When = n (8) (9) e lowe uppe ML pedcton bound w cove of ae obtaned fo e nuecal oluton of e followng equaton: Ingaa L ; (3) Ingaa U ; (3) The ML pont pedcton of e lowe ecod value gven fo (5) a f d ML q exp qd V (3) One can ue a nuecal ntegaton technque to get e above ntegaton gven by (3) 5 Bayean Pedcton fo Ode tattc Ode tattc ae n any pactcal tuaton a well a e elablty of yte It well-nown at a yte called a -out-of- yte f t cont of coponent functonng atfactoly povded at at leat coponent functon If e lfete of e coponent ae ndependently dtbuted en e lfete of e yte concde w at of e ode tattc fo e undelyng dtbuton Theefoe ode tattc play a ey ole n tudyng e lfete of uch yte ee Anold et al [6] Davd Nagaaja [7] fo oe detal concenng e applcaton of ode tattc uppoe n ndependent te ae put on a tet e lfete dtbuton of each te gven by () Let X X X3 X be e odeed -falue obeved unde e type-ii pogevely cenong plan w bnoal eoval R R at be a econd ndependent o aple (of ze ) of futue ode tattc obeved fo e ae dtbuton Ou a to obtan Bayean pedcton about oe functon of Let be e odeed lfete n e futue aple of lfete The denty functon of fo gven of e fo h D F F f (33) D Fo e Bu-X odel ubt- tutng () () n () we obtan h D exp U l l l l (34) Theefoe fo (9) (34) e Baye pedctve denty functon of wll be (ee Equaton (35)) l l D U l l x f h x d l log G q l U (35) Copyght 3 cre

5 66 A A OLIMAN ET AL ae gven by Equaton () The Bayean pedcton bound fo e futue ae obtaned by evaluatng P x fo (35) at fo oe gven value of It follow P x f x d log exp l G q q l l (36) The pedctve bound of a two-ded nteval w cove % fo e futue ode tattc ay u obtaned by olvng e followng two equaton fo e lowe LL uppe UU bound: l G q q l log exp LL l l G q q l log exp UU l (37) (38) Now by ung (35) e Bayean pont pedcton of e futue ode tattc unde E (B) LINEX lo functon (BL) ae gven epectvely a f x d B l l Log exp BL c f x d c (39) (4) I q l logu d (4) I (4) exp c q l logu d We can ue a nuecal ntegaton technque to get e above ntegaton gven by (4) (4) 6 ML Pedcton fo Ode tattc In ecton e ML pedcton bo pont nteval ung plug-n pocedue fo e futue ode tattc baed on pogeve type-ii cenoed aple defned by () Fo (34) by eplacng n e denty functon of fo gven by whch we can fnd t fo e nuecal oluton of e Equaton (7) en e denty functo n of fo gven h D l l exp U l l (43) The ML pedcton bound fo e futue ae obtaned by evaluatng P x fo oe gven value of It follow fo (43) at Copyght 3 cre

6 A A OLIMAN ET AL 67 P x l l h d exp l (44) The pedctve bound of a two-ded nteval w cove fo e futue ode tattc ay u obtaned by olvng e followng two equaton fo e lowe LL uppe UU bound: l l exp LL l (45) l l UU exp l (45) Fo (43) e ML pont pedcton of e ode tattc gven by ML h d l d l exp U (46) One can ue a nuecal ntegaton technque to get e above ntegaton gven by (46) 7 Illutatve Exaple ulaton tudy Exaple : In exaple a pogeve type-ii cenoed aple w o eoval fo e Bu-X dtbuton have been geneated ung e followng algo Algo ) pecfy e value of n ) pecfy e value of 3) pecfy e value of paaete p 4) Geneate a o aple w ze fo Bu-X ot t 5) Geneate a o nube fo bo n p 6) Geneate a o nube fo bon p fo each 3 l 7) et accodng to e followng elaton n l f n l l l ow In ee aple we aued at e exact value of P ae epectvely n = = 7 e aple obtaned gven a follow X R : ( 446) ( 5449) (678) (994) (655) (368) (47) We ued e above aple to copute: ) Bayean pont pedcton unde E LINEX lo functon; ) The 95% Bayean pedcton nteval of e unobeved lowe ecod (ode tattc); 3) The axu lelhood pedcton ML; 4) The 95% axu lelhood pedcton nteval of e unobeved lowe ecod (ode tattc); 5) The eult obtaned ae gven n Table -4 Exaple : ulaton tudy In exaple we dcu eult of a ulaton tudy copang e pe- foance of e pedcton eult obtaned n pape Ftly we geneate ( = 5) lowe ecod value (ode tattc) fo e Bu-X dtbuton = 6374 By ung e geneatng data we pedct e 9% 95% Bayean pedcton nteval fo e futue obevaton lowe ecod (ode tattc) fo e e Bu-X dtbuton by epeated e geneaton te we can fnd e Pecentage (CP) we ue po equal (3) Table 5-8 how e 9% 95% Bayean (B) axu lelhood (ML) pedcton nteval fo e futue lowe ecod (ode tattc) The aple obtaned gven a follow (464) (54) (733) (893) (8665) (873) (3) (4969) 8 Concluon In pape we conde e two-aple pedct on n e obeved pogeve Type-II cenoed aple w o eoval fo e Bu-X dtbuton fo e nfoatve aple dcued how pont pedcton pedcton nteval can be contucted fo futue lowe ecod (ode tattc) Bayean ML pedcton bo e pont pedcton e pedcton nteval ae peented dcued n pape The coonly ued fequentt appoache uch a e axu lelhood etate e plug-n pocedue whch to ubttute a pont etate of e unnown paaete nto e pedctve dtbuton ae evewed dcued Nuecal exaple ung ulated data wee ued to llutate e pocedue developed hee Fnally ulaton tude ae peented to copae e pefoance of dffeent eod of pedcto n A tudy of oly geneated futue aple fo e ae dtbuton how at e actual pedcton level ae atfactoy Fo e eult we note e followng: ) The eult n Table -4 how at e leng of e pedcton nteval ung plug-n pocedue (MLPI) ae hote an at of pedcton nteval ung Baye pocedue ) The ulaton eult how at fo all cae (lowe ecod ode tattc) e popoed pedcton level ae atfactoy copaed w e actual pedcton level 9% 95% Copyght 3 cre

7 68 A A OLIMAN ET AL E Table Pont nteval BP fo e futue lowe ecod LINEX 95% BPI fo c = c = c3 = [Lowe Uppe] Leng [899866] [77347] Table Pont nteval 95% MLPI fo ML [Lowe Uppe] [3573] [337844] [596] 65 Leng 686 [39545] [8 455] [8 77] [ ] [ ] 6343 E Table 3 Pont nteval BP fo e futue ode tattc LINEX 95% BPI fo c = c = c 3 = [Lowe Uppe] Leng [777999] [45453] [ ] [647968] [ ] 585 Table 4 Pont nteval 95% MLPI fo ML [Lowe Uppe] Leng 5895 [97855] [49 973] [ ] [ ] [33 43] 399 Table 5 Two aple pedcton fo e futue lowe eco d-9% 95% BPI fo 5 e actual pe- dcton w = = 3 = 6374 p = 4 n = = 8 9% BPI fo 95% BPI fo [Lowe Uppe] Leng CP [Lowe Uppe] Leng CP [73797] [9984] [74967] [743333] [5588] 86 9 [38996] [66679] [4775] [3584] [6467] Copyght 3 cre

8 A A OLIMAN ET AL 69 Table 6 Two aple pedcton fo e futue lowe ecod-9% 95% MLP I fo 5 e actual pe- dcton w = = 3 = 6374 p = 4 n = = 8 9% MLPI fo 95% MLPI fo [L owe Uppe] Leng C P [Lowe Uppe] Leng CP [ ] [384] [ 45] [59338] [5934] [98444] [7774] [5586] [484568] [ ] Tab le 7 Two aple pedcton fo e futue ode tattc -9% 95% BPI fo 5 e actual pe- dcton w = = 3 = 6374 p = 4 n = = 8 = 5 9% BPI fo 95% BPI fo [Lowe Uppe] Leng CP [Lowe Uppe] Leng CP [ 8999] [89867] [3447] [47344] [ ] [494797] [6968] [67579] [94496] 58 9 [848335] Table 8 Two aple pedcton fo e futue ode tattc-9% 95% MLPI f o 5 e actual edcton w = = 3 = 6374 p = 4 n = = 8 = 5 p 9% MLPI fo 95% MLPI fo [Lowe Uppe] Leng CP [Lowe Uppe] Leng CP [944] [7549] [448895] [39475] [ ] [574587] [83736] [754889] [56835] [ ] ) In geneal e ulaton eult how at e plug-n pocedue (MLPI) pefo bette an e Baye eod (BPI) n e ene of hoted nteval leng 9 Acnowledgeent The auo would le to expe e an to e edto efeee fo e ueful coent ugge- ton on e ognal veon of anucpt REFERENCE [] G J Hahan an d WQ Meee tattcal Int eval: A Gude fo Pacttone Johan Wley on Hoboen 99 do:/ [] I R Dunoe The Bayean Pedctve Dtbuton n Lfe Tetng Model Technoetc Vol 6 No pp do:8/ [3] G Lngappaah Bayean Appoach to Pedcton e pacng n e Exponental Dtbuton Annal Inttute of tattc & Maeatc Vol 3 No 979 pp 39-4 [4] I G Evan A M Ng Bayean One-aple Pedcton fo e Two-Paaete Webull Dtbuton IEEE Tanecto n Vol 9 No 98 pp 4-43 [5] E K Al-Huan Z F J aheen Paaetc Pedc- ton Bound fo e Futue Medan of e Exponental Dtbuton tat tc Vol 3 No pp do:8/ [6] I G Evan A M Ng Bayean Pedcton fo e Left Tuncated Exponental Dtbuton Technoetc Vol No 98 pp -4 do:8/ [7] I G Evan A M Ng Bayean Pedcton fo Two-Paaete Webull Lfete Model Councaton n tattc Theoy & Meod Vol 9 No 6 98 Copyght 3 cre

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