Some Estimators for the Population Mean Using Auxiliary Information Under Ranked Set Sampling

Transcription

1 Jourl of Moder Appled Sttstcl Methods Volue 8 Issue Artcle Soe Esttors for the Populto Me Usg Aulr Iforto Uder ked Set Splg Wld A. Abu-Deh Sult Qboos Uverst budeh@hoo.co M. S. Ahed Sult Qboos Uverst shed@squ.edu.o. A. Ahed rouk Uverst Hsse A. Muttlk Kg Fhd Uverst of Petroleu & Merls hstt@kfup.edu.s Follo ths ddtol orks t: Prt of the Appled Sttstcs Coos Socl Behvorl Sceces Coos the Sttstcl heor Coos ecoeded Ctto Abu-Deh Wld A.; Ahed M. S.; Ahed. A.; Muttlk Hsse A. (009 "Soe Esttors for the Populto Me Usg Aulr Iforto Uder ked Set Splg" Jourl of Moder Appled Sttstcl Methods: Vol. 8 : Iss. Artcle 4. DOI: 0.37/s/ Avlble t: hs egulr Artcle s brought to ou for free ope ccess b the Ope Access Jourls t DgtlCoos@WeStte. It hs bee ccepted for cluso Jourl of Moder Appled Sttstcl Methods b uthorzed edtor of DgtlCoos@WeStte.

2 Jourl of Moder Appled Sttstcl Methods Coprght 009 JMASM Ic. M 009 Vol. 8 No /09/$95.00 Soe Esttors for the Populto Me Usg Aulr Iforto Uder ked Set Splg Wld A. Abu-Deh M.S. Ahed. A. Ahed Sult Qboos Uverst Sult Qboos Uverst rouk Uverst Hsse A. Muttlk Kg Fhd Uverst of Petroleu & Merls Aulr forto s used log th rkg forto to derve severl clsses of esttors to estte the populto e of vrble of terest bsed o SS (rked set sple. he propertes of these el suggested esttors ere eed. Coprsos betee specl cses of these esttors other ko esttors re de usg rel dt set. Soe of the e esttors re superor to the old oes ters of bs e squre error. Keords: Aulr vrbles effcec rkg rked set sple. Itroducto M uthors hve dscussed the use of suppleetr forto of ulr vrbles surve splg to prove the estg esttors (for eple Cochr 977. he rto esttor s og the ost cool dopted to estte: ( populto es or ( the totl of soe vrble of terest fro fte populto th the help of ulr vrble he the correlto coeffcet betee the to vrbles s postve. Whe the correlto coeffcet betee the to vrbles s egtve the product esttor s used. hese esttors re ore effcet.e. hve sller Wld Abu-Deh s ssocte Professor the Deprtet of Mthetcs Sttstcs t Sukt Qboos Uverst/Sultte of O. El: budeh@hoo.co. M.S. Ahed s Assocte Professor the Deprtet of Mthetcs Sttstcs t Sukt Qboos Uverst Sultte of O. El: shed@squ.edu.o.. A. Ahed s Lecturer t Alosul Uvrst/Irq. Hsse A. Muttlk s Professor the Deprtet of Mthetcs Sttstcs t Kg Fhd Uverst of Petroleu & Merls Sud Arb. El: hstt@kfup.edu.s. vrces th the usul esttors of the populto e bsed o the sple e of sple ro sple (SS. ked set splg (SS c be used he the esureet of sple uts dr fro populto of terest s ver lborous or costl but severl eleets c be esl rrged (rked the order of gtude. khs Wkoto (968 estblshed the theor of SS. he shoed tht the e of the SS s ubsed esttor for the populto e s ore effcet th the e of SS. Dell Clutter (97 studed the effect of rkg error o the effcec of SS. he SS hs sttstcl pplctos bolog evroetl studes (Brbes & El-Shr 00 for eple McItre (95 frst suggested usg SS to estte the eld of psture. I ddto SS hs bee vestgted b reserchers (Stokes 977; Stokes & Sger 988; L et l ; Mode et l. 999; Al-Sleh & Al-Shrft 00; Al-Sleh & Zheg 000; Al-Sleh & Al-Or 000 for ore detls bout SS see Kur et l he SS ethod c be surzed s follos: Select ro sples of sze uts ech rk the uts th ech sple th respect to the vrble of terest b vsul specto or soe other sple ethod. 53

3 ESIMAOS FO HE MEAN UNDE ANKED SE SAMPLING Net select for ctul esureet the th sllest ut fro the th sple for.... I ths totl of esured uts re obted oe fro ech sple. he ccle be repeted r tes to get sple of sze r. hese r uts for the SS dt. Note tht SS r eleets re detfed but ol r of the re qutfed. hus coprg ths sple th sple ro sple (SS of sze r s resoble. Soe Notos Prelres Let deote the vrble of terest hose populto e vrce re respectvel. Estte usg the forto provded b oe or to ulr vrbles bsed o SS SS ll be cosdered. Let be the populto e vrce for. Let deote the vlues of the vrbles respectvel o the th ut of the populto. he populto es of the ulr vrbles re ssued to be ko. Let ( ( ( represet the th order sttstcs of sple of sze the th ccle of the vrbles respectvel bsed o SS of sze r dr fro the populto. he sple e for ech vrble usg SS dt re defed s follos: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( the: ( ( ( ( ( ( ( ( ( ( ( ( ( ( E ( ( ( E ( ( ( E. ( ( ( ( 0 ( 0 ( 0 ( ( ( ( ( ( ( ( ( ( ( ( r r ( ( r ( ( Cosder the follog ottos:. ( (. he follog clsses of esttors of the e of the vrble bsed o SS re: ( ( ( (. 54

4 ABU-DAEH AHMED AHMED & MULAK ( ( ( + (. here re costts +. Esttors Bsed o SS Oe or o Aulr Vrbles It s ot possble to rk to or ore desol dt therefore rkg oe of the vrbles tkg the correspodg vlues of other vrbles s opto. Assug tht the vrble c be rked perfectl - there re o errors rkg the uts there ll be errors rkg the other vrbles. kg o Stud Vrble Assue tht the rkg o vrble s perfect hle the rkg o vrbles ll hve errors; the esttors (. (. re respectvel gve b: here [ ] [ ] ( (3. [ ] [ ] ( (. + r [ ] [] r [ ] [] (3. re the sple es of the SS for respectvel [] [] re the th udget order sttstc of the th sple of th the ccle of the vrbles respectvel. Let e ( 0 e [ ] [ ] e. Obt the bs the MSE of the esttors ~ ~ respectvel up to the order of s follos: ( B r ( ( + [] [] r ( + ( [] + r ( ( [] r + ( [] r (3.3 he MSE of ~ he rkg o vrble s: MSE ( ( ( [] + + r r ( [] ( [] + r r [] [] r r ( ( + + (3.4 55

5 ESIMAOS FO HE MEAN UNDE ANKED SE SAMPLING up to the order of. he optu vlues of ~ re hch ze the MSE of obted b the dervto of (3.4 th respect to respectvel [] [] ( ( ( ( ( ( [] [] [] [] ( [] (3.5 ( ( [] [] ( ( [] [] [] [] [] ( ( ( he u MSE up to ters of clss ~ s: ( (3.6 for the MSE [( ( [] r + ( ( ( [] ( ( ( ( [] ( ( ( [] (3.7 ] If tke the vlues (3.5 (3.6 respectvel the bs of ~ fro (3.3 s gve b: B ( g [] [] [] r {[ ][ ] [ ] } (3.8 here g s gve the Apped. he bs the MSE of the esttors of (3. re gve b: ( B { [ ] [] r + [ ] [] r ( + [ ] [] r ( [ ]} [] r (3.9 + up to the order of. he MSE of the esttor f rkg o vrble s: 56

6 ABU-DAEH AHMED AHMED & MULAK MSE ( ( [] { r r [ ] + + [] [] r r [ ] [ ] + + [ ] [ ] } [] [] + r r (3.0 f re both ko tke the vlues (3.5 (3.6 respectvel up to order. he optu vlues of hch ze the MSE of ~ obted b the dervto of equto (3.0 th respect to uder the restrcto + re gve b: [] [] + [] [] [] + [] [] the MSE of [ ] [ ] ( + ( (3. s the u MSE up to ters of the clss ~ : MSE (. As for + ( [] r + [] [] [] [] + [] [] + [] + [] (3.3 If tkes the vlue (3. the bs of ~ fro (3.9 s gve b: B ( [] [] ( ( [] [] ( [] [] (3.4 kg o Oe Aulr Vrble If the rkg of s perfect the the to esttors (. (. re gve b: ( [ ] [ ] (3.5 57

7 ESIMAOS FO HE MEAN UNDE ANKED SE SAMPLING ( [ ] [ ] + ( (3.6 he foruls for the bs MSE of esttors (3.5 (3.6 respectvel ll be the se s 3. ecept for the curret esttors replce [] b( ( b []. Slrl f the rkg o s perfect the the esttors: ( [ ] [ ] (3.7 ( [ ] [ ] + ( (3.8 result. he foruls for the bs the MSE of esttors (3.7 (3.8 respectvel ll be the se s 3. ecept for the cse of replcg [] b( ( b [] ( 0 or 0 (. 0 or 0 correspod to the cse of oe ulr vrble. Coprsos of Esttors Cosder the follog ko esttors. he SS sple e of the dt: ( ( r s ubsed esttor for the populto e ts vrce s gve b: Vr ( ( [ ] ( r (khs & Wkoto 968. he to esttor usg SS dt s defed s: (. [ ] hs esttor s specl cse of the esttor equto ( here 0. he bs the MSE of ths esttor re respectvel gve b: ( B r ( [] [] MSE + ( [] r [] (S & Muttlk 996. he product esttor usg SS dt s defed s: P ( [ ] hs esttor s specl cse for the esttor equto ( here 0. he e esttor s clled the product esttor ts bs MSE respectvel re gve b B ( P [ ] [] r 58

8 ABU-DAEH AHMED AHMED & MULAK MSE ( P { [ ] + ( r [ ] [ ]} + [] [] If s set the esttor of equto ( the follog e esttor results: ( [ ] [ ] he bs the MSE re respectvel gve b B ( [] r + + [] [] } [] [] MSE ( ( r + + [] [] [] [] [] + Settg the esttor of equto ( results e esttor defed s: [ ] [ ] (.. he bs the MSE re respectfull gve b ( B + [] r + [] [] MSE r ( ( If the esttor of equto 3 s set e esttor clled the Multvrte rto esttor usg SS c be defed s ( [ ] [ ] + he bs the MSE re respectvel gve b. B ( [] + [] 59

9 ESIMAOS FO HE MEAN UNDE ANKED SE SAMPLING MSE ( ( r + + [] [] [] [] + [] [] [] +. [] [] he coprso betee the esttors proposed s llustrted b usg rel dt set. he dt for the llustrto s tke fro Ahed (995; the populto cossts of 33 vllges. Cosder the vrbles here s uber of cultvtors s the re of the vllge s the uber of household the vllge. he follog steps surze the sulto procedure to fd the bs MSE of esttor for the populto e usg perfect rkg o the vrble of terest. Step : Sulte r observtos fro the 33 rel dt vlues th replceet perfor the SS procedure th 5 r 6 to get sple of sze r 80. Step : Use the dt Step to clculte r ( ( : ( : r here (: s the sze 5 the ccle. th th sllest the sple of Step 3: epet steps (30000 tes usg these vlues to obt ( ( Step 4: Fd the pprote bs MSE for Ŷ (. he bs s obted b B ( ˆ ( ˆ ( the MSE of ( ˆ s obted s 30000( ( ˆ ( ˆ ˆ ( ( MSE he bove sulto s prefored for ll other esttors suggested rkg o oe of the vrbles or. Clculte the effcec of these esttors th respect to the MSE ( Vr( ( esttor usg ( ( ( ( MSE e ( MSE ˆ here represets of the esttors gve. I bles -3 MSE bs effcec hve bee clculted for ech of the suggested esttors. I ble rkg o the vrble s sho (.e. the rkg of vrble ll be perfect hle the rkg of the other vrbles ll be th errors rkg. bles 3 sho the rkg o the vrbles respectvel. Cosderg the results of bles -3 t s observed tht ~ dotes ll other esttors cheved the hghest effcec. Its effcec s ore th tes hgher th the 60

10 ABU-DAEH AHMED AHMED & MULAK ble : he Bs MSE the Effcec for ll Esttors Bsed o kg of the Vrble Esttor Aulr vrble MSE Effcec Bs ( Noe ~ ~ ~ ~ P P

11 ESIMAOS FO HE MEAN UNDE ANKED SE SAMPLING ble : he Bs MSE the Effcec for ll Esttors Bsed o kg of the Vrble Esttor Aulr vrble MSE Effcec Bs [ ] Noe ~ ~ ~ ~ P P

12 ABU-DAEH AHMED AHMED & MULAK ble 3: he Bs MSE the Effcec for ll Esttors Bsed o kg of the Vrble Esttor Aulr Vrble MSE Effcec Bs [ ] Noe ~ ~ ~ ~ P P SS esttor. Soe other esttors cheved hgher effcec th ( theses esttors re: ~ ~. he esttors cheved bout the se effcec o tter hch vrble s rked o. hs provdes greter fleblt choosg the vrble to rk o sce soe of the vrbles re ore dffcult to rk th others. efereces Ahed M. S. (995 Soe Estto Procedure Usg Multvrte Aulr Iforto Sple Surves (Ph. D. hess Deprtet of Sttstcs & Opertos eserch Algrh Musl Uverst Id. Al-Sleh M. Fr & Al-Shrft K. (00. Estto of verge lk eld usg rked set splg. Evroetrcs Al-Sleh M. Fr & Al-Or Aer. (00. Mult-Stge SS. Jourl of Sttstcl Plg Iferece Al-Sleh M. Fr Zheg Gg. (00. Estto of Bvrte Chrcterstcs Usg ked Set Splg. he Austrl & Ne Zel Jourl of Sttstcs Brbes L. El-Shr A. (00. he effcec of rked set splg for preter estto. Sttstcs Probblt Letters Cochr W. G. (977 Splg techques 3rd edto (Joh Wl N. 63

13 ESIMAOS FO HE MEAN UNDE ANKED SE SAMPLING Dell D.. Clutter J. L. (97 ked set splg theor th order sttstcs bckgroud Boetrcs Kur A. Ptl G. P. Sh B. K. lle C. (995 ked set splg: otted bblogrph Evroetl Ecologcl Sttstcs L K. Sh B.K. Wu Z.(994. Estto of preters to-preter Epoetl dstrbuto usg rked set splg Als of the Isttute of Sttstcl Mthetcs 46( McItre G. A. (95 A ethod of ubsed selectve splg usg rked sets Austrl Jourl of Agrculturl eserch Mode N. Coquest L. & Mrker D. (999. ked set splg for ecologcl reserch: Accoutg for the totl cost of splg. Evroetrcs S H. M. & Muttlk H. A. (996 Estto of rto usg rk set splg Boetrcl Jourl Stokes S. L. & Sger.(988. Chrcterzto of rked set sple th pplcto to esttg dstrbuto fuctos. Jourl of the Aerc Sttstcl Assocto Stokes S. L. (977: ked set splg th cocott vrbles Couctos sttstcs A khs K. & Wkoto K. (968 O ubsed esttes of the populto e bsed o the sple strtfed b es of orderg Als of the Isttute of Sttstcl Mthetcs

14 ABU-DAEH AHMED AHMED & MULAK Apped g [] [] [] [] [] [] [] [] [] [] [] [] [] [] [] + + [] [] [] + [] [] [] [] [] [] [] + [] [] + [] [] [] [] + [] [] [] [] [] [] + [] [] [] + [] [] [] + [] [] [] 65