Shrinkage Estimators for Reliability Function. Mohammad Qabaha

Transcription

1 A - Najah Uv. J. es. (N. Sc.) Vol. 7 3 Shkage Esmaos fo elably Fuco مقدرات التقلص لدالة الفاعلية ohammad Qabaha محمد قبها Depame of ahemacs Faculy of Scece A-Najah Naoal Uvesy alese E-mal: mohqabha@mal.ajah.edu eceved: (8/4/) Acceped: (//) Absac A vaey of shkage mehods fo esmag ukow paamees has bee cosdeed. We deve ad compae he shkage esmaos fo he elably fuco of he wo-paamee expoeal dsbuo. Smulao expemes ae used o sudy he pefomaces of hese esmaos. Key wods: Esmao shkage elably expoeal dsbuo smulao. ملخص يه دف ه ذا البح ث ال ى ايج اد مق درات ال تقلص لدال ة الفاعلي ة للتوزي ع الاس ي ذي المعلمت ين ومن ثم المقارنة بين هذه المقدرات عن طريق المحاآاة. Ioduco I he esmao of a ukow paamee hee ofe exss some fom of a po kowledge abou he paamee whch oe would lke o ulze ode o ge a bee esmae.

2 " fo Shkage Esmaos 4 hompso (968) descbed a shkage echque fo esmag he mea of a populao. eha ad Svasa (97) poposed aohe shkage esmao fo he mea of a populao. adey ad Sgh (977) ad adey (979) descbed a shkage echque fo esmag he vaace of a omal dsbuo. Lemme (98) cosde a shkage esmao fo he paamee of he bomal dsbuo. Qabaha e. al. (986) poved ha he mea squaed eo of hompso ype esmao s smalle ha he emag shkage esmaos fo esmag he paamees of expoeal dsbuo. Qabaha (7) deve ad compae he Bayesa Shkage esmaos fo he paamees of expoeal dsbuo. I hs pape we cosde he poblem of esmag he elably fuco () of he wo-paamee expoeal dsbuo whe he po fomao egadg () s avalable he fom of a guess value. A vaey of shkage esmaos poposed by hompso (968) eha ad Svasa (97) adey (979) ad Lemme (98) ae used fo hs pupose. he pefomaces of hese esmaos ae compaed hough smulao. Esmaos cosdeed Le he legh of lfe X of a cea sysem be dsbued as f ( X; µ ) exp( ( X µ ) ) µ X >. he elably fuco of hs sysem a me s defed by () exp [- ( - µ) / ]. Le us cosde a adom sample of ems of such a sysem subjeced o es ad he es emaed as soo as he fs ( ) ems fal. Vol. 7 3 A - Najah Uv. J. es. (N. Sc.)

3 ohammad Qabaha 43 A - Najah Uv. J. es. (N. Sc.) Vol. 7 3 Le X {X () X () X () } be he fs - odeed falue mes. I s well kow fom Epse ad Sobel (954) ad Basu(964) ha () () X X X X > + µ ad () () > X ae he mmum vaace ubased esmaos fo he paamees µ ad () especvely. he vaaces of hese esmaos (see Lee (978) p63) ae gve by va ) ( > va ) ( > µ ad () ()! ) (! 4 ) va( 4 > + m u m m Fsly we cosde he hompso (968)- ype esmao fo he elably fuco ():

4 " fo Shkage Esmaos 44 (). ad () () + c( () () ) c Whee () s he guessed value of (). hompso (968) suggesed o deeme c fom () SE( ) c Whee SE () E () () I ca be show easly ha ( ) he mea squaed eo of SE ( () ) c va( () ) + ( c) ( () () ) C ( () () ) ( () () ) + va( I pacce c s esmaed by eplacg he ukow paamees by he sample esmaes. I follows ha 3 () () + ( () () ) ( () () ) + va( () ) Secodly we cosde he eha ad Svasa (97)-ype esmao. hs s gve by. Vol. 7 3 A - Najah Uv. J. es. (N. Sc.)

5 45 ohammad Qabaha () () K () () )exp b( () () ) va( () ) ( whee K ad b ae posve cosas o be suably chose such ha <K< ad b>. No geeal gudace has bee gve o how K ad b should be chose. Subsug ukow paamees by he sample esmaes we oba () () K () () )exp b( () () ) va( () ) ( I ca be vefed ha he mmum ad maxmum values of () s aaable whe b eds o ad especvely by a suable choce of K <K<. So we ake ad b () ) ( K ) va( () ) + K( () ()) Lm SE ( Lm SE ( () va( ())) b Hece fo <K< b> ad ( ) eds o ( ) SE () SE( ()). we have hdly we cosde he adey (979) - ype esmao. hs s gve by () a K () + ( K ) () K. Vol. 7 3 A - Najah Uv. J. es. (N. Sc.)

6 " fo Shkage Esmaos 46 wh k s a cosa specfed by he expemee accodg o hs belef () ad a s deemed fom SE a () I ca be show easly ha () ) ak va( () ) + [( ak ) () a( K ) ()] SE ( ad () () () a d K va( ) + d whee K + K. () d eplacg he ukow paamees by he sample esmaes we oba wh 3 () d () d () + K va( () ) ( K ) () () d K + I follows ha SE () SE( ()) eds o () s o clea ohewse. oly whe a ad () Fally we cosde he Lemme (98)-ype esmao. hs s gve by () K () + ( K ) () L k Vol. 7 3 A - Najah Uv. J. es. (N. Sc.)

7 47 ohammad Qabaha Wh K s a cosa specfed by he expemee accodg o hs belef () ad o geeal gudace has bee gve o how K should be chose. We oe ha hompso ad Lemme esmaos ae equal whe ck padey ad Lemme esmaos ae equal whe a. Compaso of Esmaos Smulao expemes ae used o sudy he pefomaces of he esmaos. A adom sample of sze fom he wo-paamee expoeal dsbuo wh µ8 ad 7 s geeaed. he veco X{X () X () X () } of he fs -odeed obsevaos s ecoded. he he mmum vaace ubased esmaos µ ad () especvely ae compued Fo a kow cosa k bewee zeo ad oe ad fo specfc values ad () ae compued. () he quaes () () () ooe Calo expemes ae epeaed 5 mes. he aveage of he 5 sample values of each squaed eo e.g. () () L ( ) s ake as a esmae of he coespodg mea squaed eo whch s deoed by SE(.). he esmaes of he mea squaed eos of () ad he elave effceces e.g. () () ) SE( () SE( ()) ) ( ae calculaed fo 3 3 k..5 b 5. Vol. 7 3 A - Najah Uv. J. es. (N. Sc.)

8 " fo Shkage Esmaos 48 Coclusos Alhough he esuls deved he followg able apply scly o lmed cases hey ae suggesve of some geeal coclusos egadg he elave effceces of he vaous mehods. We oe fom he able ha he SE of () s always smalle ha ha of ohe esmaos. I s obvous ha () () L ad () have smalle meas squaed eo ha he mmum vaace ubased esmao of (). he mea squaed eo of () s always hghe ha he mmum vaace ubased esmao of (). he advaages of () ad () ae mos maked whe s small. L Vol. 7 3 A - Najah Uv. J. es. (N. Sc.)

9 49 ohammad Qabaha able: elave effceces of he vaous shkage esmaos of elably fuco (). Sample sze 3 µ µ () o().49 No. of falu es.v. U.E of () () ()) ( K.b K. 5b 5 ( () ()) K. K. 5 ( () ()) K. K.5 ( () ()) L efeeces - Basu A.. (964). Esmaes of elably fo Some Dsbuos Useful Lfe esg. echomecs. (6) Epse B. & Sobel. (954). Some heoems eleva o Lfe esg Fom a Expoeal Dsbuo. A. ah. Sas. (5) Lee J.B. (978). "Sascal Aalyss of elably ad Lfe-esg odels". acel Dekke Ic. Newyok Lemme H. H. (98). Fom Oday o Bayesa Shkage Esmaos. Souh Afca Sas. J. (5) eha J. S. & Svasa. (97). Esmao of he ea by Shkage o a o. J. Ame. Sas. Asso. (66) adey B.N. (979). O Shkage Esmao of Nomal opulao Vaace. Commu. Sas. (8) adey B.N. & Sgh J. (977). Esmao of Vaece of Nomal opulao o Ifomao. J. Ida Assoc. (5). 4-5 Vol. 7 3 A - Najah Uv. J. es. (N. Sc.)

10 " fo Shkage Esmaos 5 - Qabaha.A.Q. (7). Bayesa Shkage Esmao. A-Najah Uvesy Joual fo eseach. A. () Qabaha.A.Q. & Abusalh.S. & Al.A. (986). O Some Shkage echques fo Esmag he aamee of Expoeal Dsbuo. Qaa Uv. Sc. Bull. (6) hompso J.. (968). Some Shkage echques fo Esmag he ea. J. Ame. Sas. Assoc. (63) Vol. 7 3 A - Najah Uv. J. es. (N. Sc.)