Stratified Extreme Ranked Set Sample With Application To Ratio Estimators

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Journl of Modern Appled Sttstcl Metods Volume 3 Issue Artcle 5--004 Strtfed Extreme Rned Set Smple Wt Applcton To Rto Estmtors Hn M. Smw Sultn Qboos Unversty, smw@squ.edu.om t J.

Sed, t J. (004 "Strtfed Extreme Rned Set Smple Wt Applcton To Rto Estmtors," Journl of Modern Appled Sttstcl Metods: Vol. 3 : Iss., Artcle. DOI: 0.37/jmsm/083370 Avlble t: ttp://dgtlcommons.wyne.

1 Journl of Modern Appled Sttstcl Metods Volume 3 Issue Artcle Strtfed Extreme Rned Set Smple Wt Applcton To Rto Estmtors Hn M. Smw Sultn Qboos Unversty, smw@squ.edu.om t J. Sed Sultn Qboos Unversty Follow ts ddtonl wors t: ttp://dgtlcommons.wyne.edu/jmsm Prt of te Appled Sttstcs Commons, Socl Bevorl Scences Commons, te Sttstcl Teory Commons Recommended Ctton Smw, Hn M. Sed, t J. (004 "Strtfed Extreme Rned Set Smple Wt Applcton To Rto Estmtors," Journl of Modern Appled Sttstcl Metods: Vol. 3 : Iss., Artcle. DOI: 0.37/jmsm/ Avlble t: ttp://dgtlcommons.wyne.edu/jmsm/vol3/ss/ Ts Regulr Artcle s brougt to you for free open ccess by te Open Access Journls t DgtlCommons@WyneStte. It s been ccepted for ncluson n Journl of Modern Appled Sttstcl Metods by n utorzed edtor of DgtlCommons@WyneStte.

2 Journl of Modern Appled Sttstcl Metods Copyrgt 004 JMASM, Inc. My 004, Vol. 3, No., /04/$95.00 Strtfed Extreme Rned Set Smple Wt Applcton To Rto Estmtors Hn M. Smw Deprtment of Mtemtcs & Sttstcs Sultn Qboos Unversty t J. Sed Deprtment of Mtemtcs & Sttstcs Sultn Qboos Unversty Strtfed extreme rned set smple ( s ntroduced. Te performnce of te combned seprte rto estmtes usng s nvestgted. Teoretcl smulton study re presented. Results ndcte tt usng for estmtng te rtos s more effcent tn usng strtfed smple rom smple (SSRS smple rom smple (SRS. In some cses t s more effcent tn rned set smple (RSS strtfed rned set smple (SRSS, wen te underlyng dstrbuton s symmetrc. An pplcton to rel dt on te blrubn level n jundce bbes s ntroduced to llustrte te metod. Key words: Smple rom smple; strtfed rom smple; rned set smple; strtfed rned set smple; rto estmton Introducton Wen smplng unts n study cn be esly rned compred to quntfcton. McIntyre (95 proposed to use te men of unts bsed on rned set smple (RSS to estmte te populton men. RSS s conducted by selectng rom smples from te trget populton ec of sze r. Rnng ec element wtn ec set wt respect to te rom vrble of nterest. Ten n ctul mesurement s ten of te element wt te smllest rn from te frst smple. From te second smple n ctul mesurement s ten of te element wt te second smllest rn, te procedure s contnued untl te element wt te lrgest rn s cosen for ctul mesurement from te r-t smple. Tus we obtn totl of r mesured elements; one from ec ordered smple of sze r ts completed one cycle. Te cycle my be repeted m tmes untl n rm elements ve been mesured. Tese n elements form te rned set smple dt. Hn M. Smw s Assocte Professor, rmou Unversty. Contct nformton s P.O. Box 36, Sultn Qboos Unversty, Alod, 3, Sultnte of Omn. Eml: smw@squ.edu.om. Smw et l. (996 nvestgted vrety of extreme rned set smples (ERSS for estmtng te populton mens. Furtermore, Smw (996 ntroduced te prncple of strtfed rned set smplng (SRSS; to mprove te precson of estmtng te populton mens n cse of SSRS. In mny stutons te quntty tt s to be estmted from rom smple s te rto of two vrbles bot of wc vry from unt to unt. For exmple, n ouseold survey, te verge number of suts of clotes per dult mle s te qunttes of nterest. Exmples of ts nd occur frequently wen te smplng unt (te ouseold comprses group or cluster of elements (dult mles our nterest s n te populton men per element. Moreover, rto ppers n mny oter pplctons, for exmple, te rto of lons for buldng purpose to totl lons n bn or te rto of cres of wet to totl cres on frm. Also, ts metod s to obtn ncresed precson of estmtng te populton men or totl by tng dvntge of te correlton between n uxlry vrble te vrble of nterest. In te lterture, rto estmtors re used n cse of SRS s well s n cse of SSRS (for n exmple see Cocrn, 977. Also, SSRS s used n certn types of surveys becuse t combnes te conceptul smplcty of smple 7

3 STRATIFIED ETREME RANKED SET SAMPE 8 rom smple wt potentlly, sgnfcnt gns n effcency. It s convenent tecnque to use wenever, one ws to ensure tt smple s representtve of te populton lso to obtn seprte estmtes for prmeters of ec sub-domn of te populton. Tere re two metods for estmtng rtos tt re generlly used wen te smplng desgn s strtfed rom smplng, nmely te combned rto estmte te seprte rto estmte. Moreover, Smw Muttl (996 used RSS to estmte te populton rto, sowed tt t provded more effcent estmtor compred wt usng SRS. Introduce n ts rtcle s te de of strtfed extreme rned set smple (. Also, te use of te de of s proposed to mprove te precson of te two metods for estmtng te rto nmely te combned rto estmte seprte rto estmte. Moreover, studed re te propertes of tese estmtors comprng tem n dfferent stutons. ter n te rtcle te prncple of ts propertes re ntroduced. Combned seprte rto estmtors usng re ten dscussed followed by smulton study te results of te smulton ncludng n llustrton of te metods usng rel dt bout te blrubn level n jundce bbes. Metodology Rned set smple for bvrte elements A modfcton of te bove procedure used by Smw Muttl (996 for te estmton of te rto. Frst coose r ndependent smples ec of sze r of ndependent bvrte elements from te trget populton. Rn ec smple wt respect to one of te vrbles or. Suppose tt te rnng s done on te vrble. From te frst smple n ctul mesurement s ten of te element wt te smllest rn of, togeter wt te vlue of te vrble ssocted wt te smllest vlue of. From te second smple n ctul mesurement s ten ten of te element wt te second smllest rn of, togeter wt te vlue of te vrble ssocted wt te second smllest vlue of. Te procedure s contnued untl te element wt te lrgest rn of s cosen for mesurement from te r-t smple, togeter wt te vlue of te vrble ssocted wt te lrgest vlue of. Te cycle my be repeted m tmes untl n rm bvrte elements ve been mesured. Note tt we ssume tt te rnng of te vrble wll be perfect, wle te rnng of te vrble wll be wt errors n rnng, or t worst of rom order f te correlton between s close to zero. Strtfed rned set smple For te -t strtum, frst coose r ndependent smples ec of sze r of ndependent elements from te -t subpopultons,,,...,. Rn ec smple wtn ec strtum, ten use te sme smplng sceme descrbed bove to obtn ndependent RSS smples of szes r, r,, r respectvely. Note tt r r... r r. Ts complete one cycle of strtfed rned set smple. Te cycle my be repeted m tmes untl n mr elements ve been mesured (see Smw, 996. Te followng strcture for te strtfed Rned set smple s used wen te rnng on te vrble n cse of bvrte elements: For te -t cycle, te SRSS s denoted by ( ( [] ( [] Strtum : ( ( r, [ r ],,,,..., ( ( [] ( [] Strtum : ( ( r, [ r ],,,,..., ( ( [] ( [] Strtum : ( r, r,,,..., m,,,,..., Smlrly for te strtfed rned set smple wen te rnng on te vrble :

4 9 SAMAWI & SAEID ( [] ( [] ( Strtum : ( [ r ], ( r,,,,..., ( [] ( [] ( Strtum : ( [ r ], ( r,,,,..., ( [] ( [] ( Strtum : ( r, r were,,..., m.,,,,..., Extreme Rned Set Smple Te extreme rned set smple ERSSs nvestgted by Smw et l. (996. Te procedure nvolves romly drwng r sets of r unts ec, from te nfnte populton for wc te men s to be estmted. It s ssumed tt te lowest or te lrgest unts of ts set cn be detected vsully or wt lttle cost. For sure, ts s smple prctcl process. From te fst set of r unts te lowest rned unt s mesured. From te second set of r unts te lrgest rned unt s mesured. From te trd set of r unts te lowest rned unt s mesured, so on. In ts wy we obtn te frst (r- mesured unts usng te frst (r - sets. Te coce of te r- t unt from te r-t (.e te lst set depends on weter r s even or odd. If r s even te lrgest rned unt s mesured. ERSS wll denote suc smple. If r s odd ten two optons exst: b For te mesure of te r-t unt we te te verge of te mesures of te lowest te lrgest unts n te r-t set. ERSSb wll denote suc smple. c For te mesure of te r-t unt we te te mesure of te medn. ERSSc wll denote suc smple. Note tt te coce (c wll be more dffcult n pplcton tn te coce ( (b. Strtfed Extreme Rned Set Smple Suppose tt te populton dvded nto mutully exclusve exustve strt, wt subpopulton sze N, N,..., N. Troug ts rtcle t lrge subpopulton symmetrc underlyng dstrbuton wll be ssumed. Te followng nottons results wll be ntroduced for ts pper. For ll,,..., r,,...,. ( W et E (, Vr(, j ( E (, Vr j ( ( ( N n r (proportonl llocton. N n r et,,..., * * *,,..., * * * r ; ;...; * * *, r,..., r r r be r ndependent smples of sze r, ec ten from te t strtum (,,...,. Assume tt ec element * j n te smple s te sme dstrbuton functon F ( x wt men vrnce. For smplcty of notton, we wll ssume tt j denotes te quntttve mesure of te unt * j. Ten, ccordng to our descrpton,,..., r s te SRS from te * * * t strtum. et (,,..., ( r be te ordered sttstcs of te t smple * * *,,..., r, (,,..., r, ten from te t strtum. If r s even ten (, ( r, 3(,..., { r }(, r ( r denotes te ERSS for te t strtum. If

5 STRATIFIED ETREME RANKED SET SAMPE 0 r s odd ten,, (,..., { }, r r r r ( 3 r denotes te ERSS c for te t strtum. Note tt ts wll be repeted for ec (,,...,. Te resultng ndependent ERSSs from ec strtum wll be denotes te strtfed extreme rned set smple. Ts process cn be repeted m ndependent tmes. Estmte of Populton Men Usng To estmte te men usng of sze n, ssume tt tere s ( strt wt even set sze (- strt wt odd set sze. For smplcty of notton, let m ten n r n r, ten te estmte of te men usng s gven by c W W ( c, were ( ( ( r, ( (... r { r } r, 3 r r r r { } ( r ( r ( r. r r It cn be sown tt (Smw et l., 996. ( ( ( E r r E r r r ( c ( ( ( r. Terefore, te men vrnce of re E ( W ( ( ( r W Vr ( ( r r r ( ( ( r W W r ( ( r r r,. Note tt te elements n re ndependent so re te elements n ( c. Furtermore, te elements n re ndependent of te elements n ( c so re ndependent of te element n. If te underlyng dstrbuton for r ec strtum s symmetrc ten t cn be ~ sown tt E ( (.e., n unbsed estmtor Vr ( W W ( r ( r r (. Note tt te estmte of te men usng SSRS of sze r s gven by SSRS W. Also, te men vrnce of SSRS re nown to be E ( SSRS (.e., n unbsed estmtor

6 SAMAWI & SAEID Vr ( SSRS r W (. (see Cocrn,977. Teorem: Assume tt te underlyng dstrbuton for ec strtum follows Norml or ogstc dstrbuton. Ten ( Vr( Vr. SSRS Proof: Assume lrge subpopulton szes ( N, N,, N. In cse of Norml, or ogstc dstrbuton functons te followng re true, (,..., f r s even r,..., r f r ( s odd. Also note te (,,,..., r (Arnold, 99. By comprng (. (., snce (, r ( r r W r W ( 0, terefore ( 0, terefore W ( ( r r r, Smulton Study Te norml logstc dstrbuton s used n te smulton. Smple sze r 0, 0 number of strt 3 re consdered. For ec of te possble combnton of dstrbuton, smple sze dfferent coce of prmeters 000 dt sets were generted. Te reltve effcences of te estmte of te populton men usng wt respect to SSRS, SRS, RSS re obtned. Te vlues obtned by smulton re gven n Tble. Our Smulton ndctes tt estmtng te populton mens usng s more effcent tn usng SSRS or SRS. In some cse, wen te underlng dstrbuton s norml wt (.0, 3.0, 3 5.0, r 0, te smulton ndctes tt estmtng te populton men usng s even more effcent tn usng RSS, of te sme sze. Seprte Rto Estmton usng In ts Secton, obtn te seprte rto estmtor ws obtned usng strtfed extreme rned set smple. Also, te symptotc men vrnce of te estmtor were derved. Two cses re consdered, te frst cse f te rnng on vrble s perfect, wle te rnng of te vrble wll be wt errors n rnng. Te second cse, wen te rnng on vrble s perfect, wle te rnng of te vrble wll be wt errors n rnng. Also, some comprsons of te two cses re nvestgted. ( Vr( ence Vr. SSRS

7 STRATIFIED ETREME RANKED SET SAMPE Tble. Te reltve effcency of te smulton results. Dstrbuton functon n RE ( SSRS, SSRSS RE RSS, RE SRS, ( ( Norml W.3, W 0.3, W 0 3.0, 3.0, ,.0, 3.0 Norml W.3, W 0.3, W 0 3.0,.0, 3.0,.0, Norml W.3, W 0.3, W 0 3.0, 3.0, ,., 3. ogtc W.3, W 0.3, W 0 3.0, 3.0, ,.0, 3.0 ogtc W.3, W 0.3, W 0 3.0,.0, ,.0, 3.0 ogtc W.3, W 0.3, W 0 3.0, 3.0, 3.0,.,

8 3 SAMAWI & SAEID Rto Estmton wen rnng on Vrble. Assumng tt we cn only rn on te vrble so tt te rnng of wll be perfect wle te rnng of wll be wt error n rnng. If r s even ten ( [], (,(,, r r ( 3 [] 3 ( ( { }[] r { }( r (, r r r r,,...,,,, denotes te for te -t strtum. If r s odd ten ( [], (,(,, r r ( 3[] 3( ( { r } { } r r r,,...,,,, r r r r denotes te c for te -t strtum,,,..., m. Te seprte rto estmte requres nowledge of te strtum totls η n order to be used for estmtng te populton men or totl. Ten usng te sme notton of Secton (. of te wen rnng on vrble, ten te rto cn be estmted wtn ec strtum s follows: ˆ f (r s even R were R c ˆ ( c [ c] r f (r s odd, m [] r { }[], mr m [ r ] [ r ], mr m c r { [] 3 []... r, { } r r } r r mr ( ( ( r m { } (, mr r m ( r ( r mr r n mr, c m { ( 3 (... r { r }[ r] r } r mr, Note tt te smple szes re dfferent from one strtum to noter. Terefore, ssume wtout loss of generlty tt te frst ( strt ve even set sze ( r,,,...,, te lst (- strt ve odd set sze ( r,,,...,. Ts mples tt, te seprte rto estmtor usng strtfed extreme rned set smple wen te rnng on vrble, wll be s follows: R ˆ η η η η, (3. c c

9 STRATIFIED ETREME RANKED SET SAMPE 4 R ˆ W W were N W, N (nown. c η N c, (3. η N It cn be sown usng te Tylor seres expnson metod tt ( mr mn E( Rˆ O. Also, te pproxmte vrnce of R ˆ cn be obtn s follows: Snce we ve ndependent strt te ssumpton of symmetrc mrgnl dstrbuton, ten Vr( Rˆ c Vr( W W c Vr( Rˆ W ( ˆ V R ERSS W ( ˆ V R ERSSc (3.3 Usng smlr rgument s n Smw Muttl (996, we ve Vr( Rˆ R [] ( [] ( ( rm ERSS (3.4 ( r [] ( r r r (3.5 Terefore, te pproxmte vrnce of seprte rto estmtor usng (rnng on vrble s Vr ( Rˆ ERSSc ( r [] [] r r ( r ( r r R ( n Vr( Rˆ ( R [] ( [] ( W n r [] R W ( n r r r ( r r r [] (, (3.6 ( E were E ( E ( ( ( E [] [] [],

10 5 SAMAWI & SAEID E r r r r r E E r r r E [] ( E ( ( [] [] R. Rto Estmton wen Rnng on Vrble. Smlrly by cngng te notton of perfect rnng, by mperfect rnng, for. Also, by usng te sme notton of te wen rnng on vrble, ten te seprte rto estmtor usng wen te rnng on vrble, wll be s follows: R ˆ W W c In te sme wy s n secton (3. we get te followng results: Vr( Rˆ R ( [] ( [] W ( n r R W ( n r r [] r r r r ( [] c (, (3.7. (3.8 Rnng on wc vrble? Agn, snce one cn not rn on bot vrbles t te sme tme some tme t s eser to rn on one vrble tn te oter, ten we need to decde on wc vrble we sould rn. We need to compre te vrnce of Rˆ n ( 3.6 vrnce Rˆ Teorem 3. : Assume tt tere re lner reltons between,.e., ρ > 0 t s esy to rn on vrble. Also ssume tt te pproxmton to te vrnce of te rto estmtors Rˆ Rˆ gven n equtons ( 3.6 ( 3.8 respectvely re vld te bs of te estmtors cn be gnored. If underlyng dstrbuton re Norml or ogstc dstrbuton, ten Vr( Rˆ Vr( Rˆ. Proof : To prove te bove we consder smple lner regresson model between, & ec s eter Norml or ogstc mrgnl dstrbuton functon. α β ε, (3.9 α β ( 3.0 were α β re prmeters ε s rom error wt E( ε 0, Vr( ε Cov( ε, ε j 0 for j,,,..., r, lso ε re ndependent. et ( f te rnng of te -t order sttstc n te -t smple s correct f te rnng of te -t order sttstc n te -t smple s not correct.e., rdom order Note tt, ccordng to our defnton by te ssumpton of te underlyng dstrbutons

11 STRATIFIED ETREME RANKED SET SAMPE 6 (Arnold, 99. ( [] Cse. If we re rnng on te vrble we get te followng model from equton (3.9 α β ε, (3. ( [ ] [ ] were ε [] s rom error wt E( ε [ ] 0, Vr( ε [ ] Cov( ε [ ], ε [ j] 0 for j,,,..., r lso ε [] [] re ndependent. Te expected vlue of ( cn be wrtten s ( α β [ ]. (3. Te vrnce of ( s ( β []. (3.3 Now, by subtrctng ( from equton (3. multply bot sdes by ( [] [], ten te te expected vlue for te bot sdes we get, β (3.4 ( [] [] Cse. If we re rnng on te vrble we get te followng model from equton ( 3.9 α β ε. (3.5 [ ] ( [ ] Te expected vlue of [] s [ ] α β (. (3.6 Smlrly we cn sow tt [] β (. (3.7 [] ( β ( (3.8 Now, equtons (3.6 (3.8 cn be wrten s : Vr( Rˆ R ( β W [] n ( β R W r mr } x Vr( Rˆ {[ [] ] R ( β W ( n ( β R W r mr } x respectvely, terefore {[ ( ] Vr( Rˆ Vr( Rˆ. Fnlly n ts cse t s recommended to rn on vrble tt wll be used n te denomntor of te rto estmtor f we ws to estmte te men or te totl of te populton usng te rto estmtor metod wen te dt s selected ccordng metod. Combned Rto Estmton usng In ts Secton, combned were rto estmtor usng strtfed extreme rned set smple. Two cses were consdered, te frst s to me te rnng on vrble perfect, wle te rnng on te vrble wll be wt errors n rnng. Te second cse, wen te rnng on vrble s perfect, wle te rnng on te vrble wll be wt errors n rnng. Also, te propertes of tese estmtors wll be dscussed.

12 7 SAMAWI & SAEID Usng s descrbe n Secton, wen rnng on vrble. Te combned Rto estmte s defned by: were R ˆ ( ( s, (4. ( W W c [ ] W W c Terefore,, ( c ˆ. R W W [ c] W W. (4. For fxed r, ssume tt we ve fnte second moments for. Snce te rto s functon of te mens of,.e., R, ence R s t lest two bounded dervtons of ll types n some negborood of (, provded tt 0. Ten, ssumng lrge m, we cn use te Multvrte Tylor Seres Expnson, to pproxmte te vrnce get te order of te bs of te rto estmtor. Terefore, Vr( Rˆ R W [] ( [] ( mr ( ( r ( r [] r W R ( mr r ( r r ( r [] ( r r. (4.3 r Rto Estmton wen Rnng on vrble. Smlrly, te estmte s gven by: ˆ [ ] R, (4.4 ( were ( W W c W W [ c] [ ] Terefore, n combned cse, we get: Rˆ W c W c W W (4.5 Usng te sme rgument s n secton (4., ( ˆ E R R O( mn ( mr, S

13 STRATIFIED ETREME RANKED SET SAMPE 8 Vr( Rˆ ( [] ( [] ( W R mr R ( r W ( mr r r r ( r [] ( r ( r [] ( r r. r (4.6 Rnng on wc vrble? Agn, snce we cn not rn on bot vrbles t te sme tme some tme t s eser to rn on one vrble tn te oter, ten we need to decde on wc vrble we sould rn. We need to compre te vrnce of Rˆ n (4.3 vrnce of Rˆ n (4.6. Teorem 4. : Assume tt tere re lner reltons between,.e., ρ > 0 t s esy to rn on vrble. Also ssume tt te pproxmton to te vrnce of te rto estmtors Rˆ Rˆ gven n equtons (4.8 (4. respectvely re vld te bs of te estmtors cn be gnored, f underlyng dstrbuton s Norml or ogstc dstrbuton, ten Vr( Rˆ Vr( Rˆ Proof: Te proof s smlr to tt of Teorem 3.. Fnlly n ts cse t s lwys recommended to rn on vrble tt wll be used n te denomntor of te rto estmtor f we ws to estmte te men or totl of te populton usng te rto estmtor metod wen te dt s selected ccordng metod. Smulton Study Computer smulton ws conducted to gn nsgt n te propertes of te rto estmtor. Bvrte rom observtons were generted from bvrte norml dstrbuton,,, correlton coeffcent ρ. Also we devled te dt nto tree strt n some cses nto four strt. Te smplng metods descrbed bove re used to drw, SRSS SSRS wt sets of sze r. We repet ts process m tmes wt prmeters to get smples of sze n rm. Te smulton ws performed wt r 0,, wt m 0 for te, SRSS SSRS dt sets. Te rto of te populton mens ws estmted from tese smples. Usng 000 replctons, estmtes of te mens, men squre errors were computed. Te rnng ws consdered on eter vrble or.e., te rnng n one of te two vrbles would be perfect wle te second wt errors n rnng. Results of tese smultons re summrzed by te reltve effcences of te estmtors of te populton rto by te bs of estmton for dfferent vlues of te correlton coeffcent ρ. Introduced ere s only one tble for effcency wen rnng on one for te bs, snce oter tbles gve te sme concluson Results of te smulton s gven n Tble for te effcency wen rnng on vrble. Tble 3 sows te bs of te estmtors wen rnng on te vrble. Te effcency of te rto estmtor s defned by MSE( R SSRS eff ( RSSRS, R. MSE( R Results It s concluded tt te gest gn n effcency s obtned by rnng of te vrble wt lrge vlues of negtve ρ. For exmple n Tble, eff ( R SSRS, R wen ( ρ.90, r & m0 s.69 wle wen eff ( R SSRS, R (ρ -.90, r & m0 s Also, our smulton ndctes te followng:

14 9 SAMAWI & SAEID Tble. Effcency wen rnng on vrble W :.3 /.3 /.4 x : / 3 / 4 y : 3 / 4 / 6 x : / / y : / / R.45 Eff. In Combne Eff. n Seprte ρ r SSRS SRSS SSRS SRSS

15 STRATIFIED ETREME RANKED SET SAMPE. Wen rnng on vrble, te effcency wll decrese wt decresng te vlue of ρ from 0.99 to 0.50, strt to ncrese s ρ decreses from 0.5 to Te effcency wll ncrese wen te even smple sze ncresed by ncresng te number of elements n ec set (r. 3. Tere wll be no cnge n te effcency f te smple sze ncresed by ncresng te cycle sze m. 4. For fx ρ, we noted tt n combned cse, s r ncrese te effcency wll ncrese, for ll vlues of ρ postve or negtve except, n some cses wen (r ρ postve. 5. Also, for fx ρ, we note tt n seprte cse, s r ncrese te effcency wll ncrese, for ll vlues of ρ postve or negtve, except n some cses wen (r ρ postve. 6. For fx r, we note tt n combned cse te effcency wll decrese from 0.99 to 0.50, ten fter ts wll ncrese from 0.5 to Also, for fx r cnge ρ, we note tt n seprte cse te effcency wll decrese from 0.99 to 0.50, ten fter ts wll ncrese from 0.5 to We note tt te effcency n combned cse less tn n seprte cse. Tt s becuse te smple sze wtn ec strtum s smll. 9. Te bs wll decrese wen ncresng te number of ncreses te even smple sze r elements n ec set. 0. Te bs n combned cse s less tn te correspondng bs n seprte cse. Applcton: Blrubn evel n Jundce Bbes Introduced s rel lfe exmple bout Blrubn level n jundce bbes wo sty n neontl ntensve cre. Most of brt surveys on lve newborns Brt sowed tt jundce s common. Jundce n new Born cn be ptologcl pysologcl wc strt on second dy of lfe t s reltonsp wt rce, metod of feedng Gesttonl ge. Jundce s observed durng te frst wee of lfe, neontl jundce s common problem. It s possble tt te generlly ccepted levels re too g my produce some g tone erng loss. Most of neontl jundce ppers on second dy of lfe. Most of norml newborn bbes leve te osptl fter 4 ours of lfe. Terefore, te prmry concern wll be on bby s wo styng n neontl ntensve cre. Pyscns re nterestng n te jundce, ccordng to ts mportnt rs on te erng, brn det. We wll focus on te wegt blrubn level n blood (tsb for te bbes. Te dt were collected on 0 bbes, wo sty n neontl ntensve cre, n four Jordnn osptls (see Smw Al- Sgeer, 00. Te dt were dvded nto two strt, mle strtum of sze N 7 femle strtum of sze N 48. Te followng re te exct populton vlues of te dt For Mles t ws found tt:.9, 0. 75,. 97, 5.5 ρ 0.. For Femles t ws found tt:.8, 0. 64, 9. 97, 4. ρ Also, for te wole dt t ws found tt:. 87, 0. 7,. 8, 5.08 ρ Two strt exst, m r0, wc produce n r.m W 0. 6, W For 0 0 Mle : n ; r 6. For Femle: n.4 0 8; r 4. 0

16 3 SAMAWI & SAEID W : 0..3/ 0.3/ 0.4 Tble : Bs of te rto estmtors wen rnng on vrble : / 3 / 4 : / / R.45 :3 / 4 / 6 : / / ρ r Combned Combned Combned Seprte SRSS SSRS SRSS Seprte SSRS Seprte

17 STRATIFIED ETREME RANKED SET SAMPE 3 Usng & SSRS We use te metod of smplng SSRS to get te followng smples. Note tt te rnng ws on vrble (wegt.te wc s drwn s n Tble 4. Bsed on te t ws found tt ˆ.83, ˆ 9. 78, ˆ 3. 37, ˆ.74, ˆ 0. 86, ˆ 4. 69, ˆ 0.377, ˆ. 68. Also, for te SSRS t ws found: ~.58, ~ 8. 9, ~. 97, ~.89, ~ 0. 7, ~ 3. 8, ~ 0.35, ~ note tt V ˆr R Now, ( ( s V ˆr( R ( c V ˆr( R SSRS ( s V ˆr R SSRS ( c ˆr( R ( s Vˆ r ˆr ˆ V ( R ( c V r R SSRS ( c V R SSRS ( s It s cler tt ts just llustrton of te computtons only. However, stll ts concluson ndctes tt te results n Sectons 3, 4 5 re correct. eff ( R SSRS ( s, R ( s 3. 73, eff ( R SSRS ( c, R ( c 3. 3 eff ( R SSRS ( c, RSSRS ( s. 00 eff ( R ( c, R ( s. 5.

18 33 SAMAWI & SAEID Cycle Number Tble 4. Te drwn smples usng SSRS metods. Femles Mles Smple tsb Wegt tbs Wegt SSRSS References Arnold, B. C., Blrsnn, N., & Ngrj, H. N. (99. A frst course n order sttstcs. N: Jon Wley & Sons. Cocrn, W. G. (977. Smplng tecnques. 3rd edton. N: Jon Wley & Sons. McIntyre, G. A. (95. A metod of unbsed selectve smplng, usng rned sets. Austrln Journl of Agrculturl Reserc 3, Smw, H. M. (996. Strtfed rned set smple. Pstn Journl of Sttstcs,,, 9-6. Smw, H. M., Amed, M. S. & Abu- Dyye, W. (996. Estmtng te populton men usng extreme rned set smplng. Bometrcl Journl, 38, 5, Smw, H. M. Al-Sgeer, O. A. M. (00. On te estmton of te dstrbuton functon usng extreme medn rned set smplng. Bometrcl Journl, 43,, Smw, H. M., & Muttl, H. A. (996. Estmton of rto usng rn set smplng. Bometrcl Journl, 38, 6, Ts, K. & Wmoto, K. (968. On unbsed estmtes of te populton men bsed on te strtfed smplng by mens of orderng. Annls of te Insttute of Sttstcl Mtemtcs, 0, -3.

Applied Statistics Qualifier Examination

Applied Statistics Qualifier Examination Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng