Mean Value Prediction of the Biased Estimators

Transcription

1 Iteratioal Joral of Scietific ad Reearch Plicatio, Volme, Ie 7, Jly ISSN 5-5 Mea Vale Predictio of the Biaed timator Rahpal Sigh, Mohider Pal Departmet of Statitic, Govt MAM College Jamm-86 Departmet of Statitic, GCW Gadhi Nagar Jamm-8 Atract- I thi paper we formlate Stei miimax type etimator ad compare the performace propertie with other etimator i cae of mea vale predictio Whe the model i etimated y ordiary leat qare it ha ee oerved that leat qare predicted i iaed while miimax ad Stei-miimax predictor are iaed he periority coditio of the etimator have ee derived y amig error ditritio to e oormal Idex erm- Liear regreio model, Miimax etimator, Stei-miimax etimator, Mea vale predictio P I INRODUCION redictio i a importat apect of relatiohip aalyi i ay cietific tdy It ot oly reflect the adeqacy of derlyig model t alo ait i makig a itale choice amog the competig model Geerally, predictio of tdy variale i liear model are otaied either for actal vale or for average-vale t ot for oth imltaeoly Sectio- decrie the model pecificatio ad the predictor Sectio- deal with the propertie of the etimator We have derived the expreio for the predictio of mea vale of the tdy variale for miimax ad Stei-miimax etimator eparately ad their relt are preeted i the form of theorem i Sectio- I Sectio-5 comparative tdy have ee made Proof of the theorem are give i Sectio-6 II MODL SPCIFICAION AND SIMAORS Let the tre liear regreio model i () Y Where Y i a ( ) vector of oervatio o the variale to e explaied, i a ( p) matrix of oervatio o p explaatory variale, ( p) i a coefficiet, ( ) eig kow calar ad i a vector of regreio vector of ditrace with mea zero, variace ity ad meare of kewe ad krtoi are ad ( +) repectively Applicatio of leat qare to () yield the ordiary leat qare etimator a Y () which i well kow to e the et liear iaed etimator of, havig variace- co-variace matrix V () he Stei rle etimator of the regreio co-efficiet from () i give y () Y Y o i ay poitive o-tochatic calar characterizig the etimatio he MIL propoed y k ad Olma (97, 97) ad k (97) i (5) D ' W Y D C (6) Uig aove defied qatitie, we otai miimax etimator (7) W W W W W he Stei- miimax regreio coefficiet of i otaied y iterchagig y i () ad i give y (8) Y Y Uig aove defied qatitie, we otai Stei miimax etimator W W (9 wwwijrporg

2 Iteratioal Joral of Scietific ad Reearch Plicatio, Volme, Ie 7, Jly ISSN 5-5 W B B W W W W W B B W U U BU B W W W W B W W U W he Predictor o tdy the performace propertie of thee etimator for predictio prpoe, let potlate the followig predictio vector () (), are defied earlier III PROPRIS OF SIMAOR I order to tdy the propertie of the etimator, we oerve that the exact expreio for the ia vector mea qared error matrice ad rik fctio of the leat qare etimator from model () ca e eaily otaied However, it i ot o with the miimax liear etimator herefore, i order to derive the expreio for the ia vector, mea qared error matrice ad rik fctio, we itrodce the followig otatio: () P () M P Here it i eaily ee that M M : M M M M M Stittig () i (), we oerve that i a iaed etimator of with variace covariace matrix a V () o o o P Where P he ditritioal amptio regardig the ditrace doe ot have ay effect o the propertie of leat qare etimator Clearly, the miimax etimator ad Stei-miimax etimator i a iaed etimator IV PRDICIONS I thi ectio, we have coidered predictio of mea vale of the tdy variale eparately Mea Vale Predictio Whe the iteret lie i predictio of mea vale of the tdy variale, o are fod to e iaed t ad are fod to e iaed ( Y) P () o o heorem (): Whe ditrace are mall, the ia vector ad predictive rik of predictor approximatio () () ( ) are give y W PB PR W p to the order of or W wwwijrporg

3 Iteratioal Joral of Scietific ad Reearch Plicatio, Volme, Ie 7, Jly ISSN 5-5 heorem (): Whe ditrace are mall, the ia vector ad predictive rik of predictor approximatio () PB ( ) are give y p to the order of or W trb W I Be I Be W W I Be trb W W I B B) B W B) W W W W (5) PR W W trbi B B) B trb From () ad (), it i clearly ee that predictor ad are iaed hi i i cotrat with the iaede of O V A COMPARISON O comparig the rik aociated with ad miimax etimator we oerve that (5) R R OLS etimator W W By oervig the aove expreio, we fid that the predictor aed o miimax liear etimator perform etter tha the predictor aed o ordiary leat qare etimator O comparig the rik aociated with OLS ad Stei-miimax etimator, we oerve that W Rik Rik (5) W B I B B) B trb Whe the ditritio of ditrace i leptokrtic or meokrtic ie, the predictor aed o Stei-miimax etimatio procedre perform etter tha that aed o ordiary leat qare predictor Hece we fid that Stei-miimax etimator i more efficiet tha ordiary leat qare etimator O comparig the rik aociated with miimax etimator, ad Stei-miimax etimator, we oerve that (5) trbi B Rik Rik B B) B trb Agai, whe the ditritio of ditrace i leptokrtic or meokrtic, it i eay to oerve that the predictor aed o Stei miimax etimator perform etter tha aed o miimax etimator Hece we fid that Stei-miimax etimator i more efficiet the miimax etimator heorem Uig (7), we get VI PROOF OF H HORMS (6) wwwijrporg

4 Iteratioal Joral of Scietific ad Reearch Plicatio, Volme, Ie 7, Jly ISSN 5-5 W W (6) W (6) (6) We have Y (65) W W Uig (65) ad (6), we oerve that Y (66), ad defied i (6), (6), (6) repectively he relt () of ia vector aociated with theorem () ca e otaied from (67) PB Y : Now, y takig expectatio of Utilizig thee expectatio i the expreio PR Y Y (68) give the predictive rik of a tated () i theorem () heorem Uig (9) i the expreio (), we oerve that S (69) Stractig (65) from (69), we oerve that Y (6) W W (6) (6) (6) (6) B W W W B W W W B B W B W B W W W W W W he relt of ia vector aociated with theorem () ca e otaied from Y (6), Now, y takig expectatio of Utilizig thee expectatio i the expreio wwwijrporg

5 Iteratioal Joral of Scietific ad Reearch Plicatio, Volme, Ie 7, Jly 5 ISSN 5-5 (65) Y Y PR give the predictive rik of a tated (5) i theorem () RFRNCS [] Akdeiz F ad S aciralar (), More o the New Biaed etimator i Liear Regreio, Sakhya, Vol 6, B,, p -5 [] Bke,O (975), Miimax liear ridge ad hrke etimator for liear parameter, Math Operatioforchg Statitik, 66,697-7 [] Chaey, YP, iwari, R ad Gpta, R (), Propertie of Mixed Geeralized Leat Sqare timator i Liear Regreio Model, Iteratioal Joral of Mathematical Sciece,Vol, No, p 7 87 [] De, M, V Srivatava ad H oteerg ad P Wijekoo (99), Stei-rle timator der Iclio of Sperflo Variale i Liear Regreio Model, Comm Statit heory Meth, (7), 9- [5] k, J ad W Olma (97), Miimax etimatio i liear regreio model, Joral of Statitical Plaig ad Iferece, Volme 5, Ie, Page [6] oteerg, H 975, Miimax-liear etimatio ad phae mmle i a retricted liear regreio model, Math Operatioforchg Statitik, 6, 7-76 [7] oteerg, H (976), Miimax-liear ad Me-etimator i geeralized regreio, Biometriche Zeitchrift, 8, 9- [8] Srivatava, A ad Shalah (995), Predictio i Liear Regreio Model with Mearemet rror", Idia Joral of Applied coomic, Vol, No, pp - [9] Shalah (995), Performace of Stei - rle Procedre for Simltaeo Predictio of Actal ad Average Vale of Stdy Variale i Liear Regreio Model, Blleti of the Iteratioal Statitical Ititte, he Netherlad, pp 75-9 [] Zeller, A (99), Bayeia ad No Bayeia timatio Uig Balaced Lo Fctio, I Statitical Deciio heory ad Related opic ed SS Gpta ad JO Berger (Spriger Verlag), p, 77-9 AUHORS Firt Athor Rahpal Sigh, Departmet of Statitic, Govt MAM College Jamm-86, -Mail: rah_7@yahoocom, Cotact No , Sject: Statitic Secod Athor Mohider Pal, Departmet of Statitic, GCW Gadhi Nagar Jamm-8, -Mail: mohider5@ymailcom, Cotact No , Sject: Statitic wwwijrporg