Regression Estimator Using Double Ranked Set Sampling

Transcription

1 Scence and Technolog 7 ( Sultan Qaboos Unvest Regesson Estmato Usng Double Ranked Set Samplng Han M. Samaw* and Eman M. Tawalbeh** *Depatment of Mathematcs and Statstcs College of Scence Sultan Qaboos Unvest P.O.Bo 36 Al Khod 3 Muscat Sultanate of Oman**Depatment of Statstcs Yamouk Unvest Ibd Jodan -63 *Emal: hsamaw@squ.edu.om. مقدر الانحدار باستخدام طريقة المعاينة المرتبة الثناي ية المراحل هاني سماوي وا يمان طوالبه خلاصة : ا داء طريقة المعاينة المرتبة الثناي ية المراحل للتقدير الانحداري سوف يبحث هنا عندما يكون الوسط الحسابي للمتغير المساعد غير معروف. ا ن التحليلات الا ولية وطرق المحاكاة دلت على ا ن مقدر الانحدار باستخدام طريقة المعاينة المرتبة الثناي ية المراحل هو ا كثر فاعلية من مقدرات الانحدار باستخدام ا ي من طرق المعاينة المرتبة وغير المرتبة الا خرى. وا يض ا هو ا فضل وا كثر فاعلية من المقدرات البسيطة باستخدام الا وساط الحسابية لا ي من طرق المعاينة المرتبة وغير المرتبة ا حادية ا و ثناي ية المراحل. ABSTRACT: The pefomance of a egesson estmato based on the double anked set sample (D scheme ntoduced b Al-Saleh and Al-Kad (000 s nvestgated when the mean of the aula vaable s unknown. Ou pma analss and smulaton ndcates that usng the D egesson estmato fo estmatng the populaton mean substantall nceases elatve effcenc compaed to usng egesson estmato based on smple andom samplng (SRS o anked set samplng ( (Yu and Lam 997 egesson estmato. Moeove the egesson estmato usng D s also moe effcent than the naïve estmatos of the populaton mean usng SRS (when the coelaton coeffcent s at least 0.4 and D fo hgh coelaton coeffcent (at least 0.9. The theo s llustated usng a eal data set of tees. KEYWORDS: Double Eteme Ranked Set Sample; Double Ranked Set sample; Eteme Ranked Set Sample; Ranked Set Sample; Regesson Estmato.. Intoducton n man applcatons consdeable cost savngs can be acheved f the numbe of quantfcatons I s onl a small facton of the numbe of avalable unts although all unts contbute to the nfomaton content of the quantfcaton. Ranked set samplng ( s a method of samplng that can acheve ths goal. was fst ntoduced b McInte (95. It s hghl poweful and much supeo to the standad smple andom samplng (SRS fo estmatng some populaton paametes. can be appled n agcultual envonmental and human populatons. Fo eample the level of blubn n the blood of nfants can be anked vsuall b obsevng: ( Colo of the face. ( Colo of the chest. ( Colo of lowe pat of the bod. (v Colo of temnal pats of the whole bod. As the ellowsh goes fom ( to (v the level of blubn n the blood goes hghe (see Samaw and Al-Sakee 00. Al-Saleh and Al-Kad (000 showed that the effcenc of estmatng the populaton mean could be mpoved even moe b usng double anked set samplng (D. Also the poved that ankng n the second stage s ease than n the fst stage. Moeove as a vaaton of Samaw et al. (996 nvestgated eteme anked set sample (E and also suggested double eteme 3

2 HANI M. SAMAWI and EMAN M. TAWALBEH anked set samplng (DE Samaw (00. Moe detals about can be found n Kau et al. (995 and Patl et al. (999. In ths pape we nvestgate the pefomance of D fo estmatng the populaton mean usng the egesson estmato. Theoetcal and numecal compasons wth othe estmatos wll be consdeed. In secton notatons defntons and some basc esults ae ntoduced. The egesson estmato usng SRS and egesson estmato (Yu and Lam 997 ae ntoduced n secton 3. Ou poposed egesson estmato usng D and ts popetes ae gven n secton 4. In secton 5 we llustate the theo usng a set of data epesentng a eal lfe stuaton.. Sample Notaton and Defnton wth Some Useful Results. One Stage Samplng.. Unvaate Populaton nvolves selectng andom sets each of se fom the taget populaton. In the most pactcal stuatons the se wll be 3 o 4. Rank each set b a sutable method of ankng fo eample b usng po nfomaton o vsual nspecton. In samplng notaton ths mples: ( ( ( afte ankng ( ( ( ( ( whee j denotes the -th obsevaton n the j -th set and j ( s the -th odeed statstc n the j -th set. Onl the elements ( (... ( ae quantfed.e. the element wth smallest ank fom the fst set the second smallest fom the second set and so on untl the lagest unt fom the -th set s measued. Ths epesents one ccle of. We can epeat the whole pocedue m tmes to get a of se n m (Takahas and Wakmoto Fo bvaate populaton Samaw and Muttlak (996 modfed the above pocedue n the case of bvaate dstbutons to estmate the populaton ato R /. The pocedue s descbed as follows: Fst choose ndependent bvaate elements fom a populaton wth bvaate dstbuton functon F(. Rank each set wth espect to one of the vaables Y o. Suppose ankng s on vaable. Appl the same pocedues as n case of unvaate populaton but fo each measued unt fom the s the assocated unt fom the Y s s measued too. Ths ma be epeated m tmes to get a bvaate sample of se n m. In sample notaton: The sample {( Y ; } wll denote the bvaate. k ( k [] k m. Double Ranked Samples (Two stage samplng As a vaaton of Al-Saleh and Al-Kad (000 ntoduced the D pocedue as follows: 3. Identf elements fom the taget populaton and dvde these elements andoml nto sets each of se elements.. Appl the usual pocedue to each set to obtan each of se. 3. Emplo agan the pocedue n Step to obtan the D of se. 4. We ma epeat steps -3 m tmes to obtan a sample of se n m. In samplng notaton afte ankng each sample sepaatel n each subset we get: Y χ (. 3

3 REGRESSION ESTIMATOR USING DOUBLE RANKED SET SAMPLING k k k ( k k k ( k k k ( k k k ( k k k ( k k k ( (. l k m whee k ( s the -th odeed obsevaton n the -th sample of the l-th set n the k-th ccle. Use scheme on each subset sepaatel to get { ( }... { } Ak. k k k (... Ak k k (. k Then n the second stage let W -th smallest obsevaton n then {W k ( k m} wll denote the D. Now let W... ( k W( k A k k ( k. m be a D whee the mean and vaance of Wk ( ae ( and ( espectvel. Al-Saleh and Al Kad (000 showed that: ( and ( ( ( + whee and ae the mean and the vaance of the populaton espectvel. Also t was shown that ankng n the second stage s ease than n the fst stage. 3. Regesson Estmatos Usng SRS and As n ato estmaton the lnea egesson estmato s used to ncease the pecson of estmatng the populaton mean b usng eta nfomaton n an aula vaable that s coelated wth the suve vaable Y. When the elaton s appomatel lnea and the lne does not go though the ogn an estmate of the populaton mean based on the lnea egesson of Y on s suggested athe than usng the ato of the two vaables. 3. Regesson estmato usng SRS Let ( Y be a bvaate sample fom whee ( F( and assume that Y + β + ε (3. ε s and ae the means of and Y espectvel and fo fed the ae..d (ndependent and dentcall dstbuted wth mean eo and vaance ε ( ρ. Consde the case whee s unknown. The method of double samplng can be used to obtan an estmate of. Ths nvolves dawng of a lage andom sample of se n whch s used to estmate. Then a subsample of se n s selected fom the ognal selected unts to stud the pma chaactestc of Y. Settng n m and n m the fst and the second-phase samples ae smple andom samples. Then the double-samplng egesson estmato Y ds s gven b Y Y + β (3. ds SRS SRS 33

4 whee m SRS HANI M. SAMAWI and EMAN M. TAWALBEH Y SRS Y s the sample mean of based on m m obsevatons of n the fst phase and β ( ( Y SRS Y SRS ( SRS When the undelng dstbuton of ( Y s assumed to be bvaate nomal the egesson estmato Yds s an unbased estmato fo and ts vaance s gven b Va Y ( ds + + ( 3 ε n n m (Sukhatme and Sukhatme 970. If the assumpton of the lnea elatonshp n (3. s nvald then the SRS egesson estmato n (3. s n geneal a based estmato of. 3. Regesson estmato usng. ρ Consde the bvaate. Fom (3. the elatonshp between Y and [] k ( k s (3.3 Y [] ( ( [] + β +ε and k m. (3.4 k k k Agan when s unknown the method of double samplng (two-phase samplng can be used to obtan an estmate of. Note that the fst-phase sample s a smple andom sample and the second-phase sample s a anked set sample. Then the double-samplng egesson estmato Y Rds based on as n Yu and Lam (997 have gven b: Y Y + β Rds (3.5 whee s the sample mean of based on the m obsevatons of the fst phase. Futhemoe usng the basc popetes of condtonal moments Yu and Lam (997 showed that Y s an unbased estmato of unde (3.4 and the vaance s gven b: whee Va Y ( Rds ( Z Z ε + E + ρ (3.6 n S R m Z ( / Z( ( / k k S R k ( Z( k Z. m k m Z Z( Agan f the assumpton of lnea elatonshp s nvald the egesson estmato n (3.5 s n geneal a based estmato of. Net we wll popose ou appoach fo usng a egesson estmato fo estmatng based on D. Rds 34

5 REGRESSION ESTIMATOR USING DOUBLE RANKED SET SAMPLING 4. Regesson Estmato Usng D 4. D fo egesson estmato In the two-phase egesson estmato usng D fo the k-th ccle n the fst stage quantfed samples each of se ae consdeed. The followng wll denote the fst stage samplng: ` { (... } { (... } { ( } A k k k k A k k k k A... k... m. k k k k These sets of quantfed obsevatons of se m ae used to estmate the populaton mean of the vaable whch s assumed to be unknown. In the second stage a bvaate D of } se nm whch s { W Y [ ] :... k... m s measued. k k Note that ankngs n the second stage on the vaable ae based on the eact measues.e. pefect ankng. Also we ae not usng the m 3 obsevaton fom the fst stage to estmate because we quantfed onl m of them and not all the m 3 obsevatons and ths wll educe the cost of the samplng unt n the stud. 4. Regesson Estmato of If W and Y ae espectvel the -th smallest value of (fom the second stage of k ( [ ] k D and the coespondng value of Y obtaned fom the - th sample n the k- th set then fom (3. we have ( ( k ( k Y + β W + εk... k... m (4. whee β s the model slope β ρ / and ε k ae andom eos as n (3.. Let be m the sample mean based on the samples of se m.e. (. k m l k Note that m ( E ( E ( ( k (4. m l k Va 3 ( (4.3 m m (see Dell and Clutte (97. Unde D the egesson estmato of the populaton mean ( ( can be defned as Y Y + ˆ β W (4.4 D D whee m ( W W Y k [ ] k ˆ m m βd k m W W Y Y k m ( W W k m k k k D [ ] k and as above. 35

6 4... Popetes of the estmato HANI M. SAMAWI and EMAN M. TAWALBEH Agan usng the basc popetes of condtonal moments and the above esults the followng theoem wll be poved. Theoem 4.: Unde (4. assumptons: ( ( E Y. ( Va ( Y ( ρ whee * ( Z + + m Z E W β m S W ( k m ( ( ( Z S Z R k Z k SS m k Z W W and Z. Poof: The pove of ths theoem and two equed popostons ae n the Append. 4.. Pefomance of Y wth Respect to Nave Estmatos Fom the pevous esults the elatve pecson of Y wth espect to the nave estmatos of Y and Y D usng and D espectvel ae as follows: ( RP Y RP Y ( gd Y ( m [ ] Re ( Z Z W β ( ρ + E + 3 m S m ( gd Y D ( ( Z ρ + + E β S ** ( Re ( Z β ( ρ + E + S ( ( ( (4.5 (4.6 Note that these elatve pecsons ae based on the vaances of the estmatos Pefomance of Y wth espect to othe Regesson Estmatos When the sample s down fom a standad bvaate nomal populaton the elatve pecson Y elatve to the two-phase (double samplng egesson estmato Y ds based on SRS (see of Sukhatme and Sukhatme 970 and to the two-phase egesson estmato Y Rds based on (see Yu and Lam 997 wll be espectvel as follows: 36

7 REGRESSION ESTIMATOR USING DOUBLE RANKED SET SAMPLING RP Y RP Y ( gd Y ds ( gd Y Rds ( ρ + + ρ m ( m 3 m ( Z ZW Re β ( ρ + E + 3 m S m ( ρ ( Z ( Z Re β ( ρ + E + 3 m S m + E ρ + m S m ( ( (4.7 (4.8 (see secton 3. Note that these elatve pecsons ae based on the vaances of the estmatos. Snce t s not eas to fnd the values of the above epessons smulaton s used to calculate them. 4.3 Smulaton Stud 4.3. Desgn of the Smulaton A compute smulaton s conducted to stud the effcenc of the egesson estmato. Usng SRS and D bvaate nomal andom samples whee geneated when 4 and ρ ± [ ]. The pefomance of the egesson estmatos ae nvestgated fo and 8 and m 4 and 8. Usng 5000 eplcatons estmates of the means and the mean squae eos fo the egesson estmatos wee computed. RE R R MSE R / MSE R The effcenc of the egesson estmato s defned b j j whee and j epesent an tpe of the above samplng methods. The esults fo the smulaton ae n Tables and. Table : The effcenc of Y wth espect to the nave estmatos based on and D m 4 8 ρ0.99 ρ 0.95 ρ 0.93 ρ 0.9 ρ 0.8 D D D D D Notce that ρ takes onl hgh postve values because the egesson estmato fo the populaton mean s used onl when the coelaton between the two vaables s hgh. Also 37

8 HANI M. SAMAWI and EMAN M. TAWALBEH negatve values ae not consdeed snce fom (4.6 and (4.7 the elatve pecson depends on the absolute values of ρ and β snce the ae squaed Results of the Smulaton Ou smulaton (Table shows that the effcenc s affected b the value of ρ. The egesson estmato based on D s moe effcent than nave estmato usng wheneve the absolute value of the coelaton coeffcent between and Y (ρ s moe that Moeove ths effcenc s nceasng as the set se o the ccle se nceases. Also the egesson estmato based on D s moe effcent than the nave estmato usng D wheneve ρ >0.90. Howeve when ρ <0.98 the effcenc deceased as the set se nceased and nceased othewse. Moeove n ths case the effcenc s not affected b the ccle se. Table shows that the double samplng egesson estmato usng D was alwas supeo to the double samplng egesson estmatos usng SRS and. Howeve the effcenc was affected b the value of ρ. The effcenc nceased b nceasng the value of ρ. Also the effcenc deceases wth nceasng the set o the ccle se fo small values of ρ. Howeve Y was stll found to be moe effcent than usng othe samplng methods. Table : The effcenc of Y wth espect to the egesson estmatos based on SRS and. M 4 8 ρ0.99 ρ 0.9 ρ 0.8 SRS SRS SRS Applcatons to Real Data Set We llustate the double anked set sample mean estmaton pocedue usng a eal data set whch conssts of the heght (Y and the damete ( at beast heght of 399 tees. See Platt et al. (988 fo a detaled descpton of the data set. The summa statstcs fo the data ae epoted n Table 3. Note that the coelaton coeffcent ρ Table 3: Summa Statstcs of tees data. Vaable Mean Vaance Heght ( n feet Damete ( n cm Populaton se N 399 and the coelaton coeffcent between and Y s ρ

9 REGRESSION ESTIMATOR USING DOUBLE RANKED SET SAMPLING Usng a set se 3 and the ccle se m3 we daw bvaate SRS and D of se 9. Table 4 contans all the above poposed estmatos and the estmated vaances usng the dawn samples. Table 4: Results fom the dawn samples Sample Naïve Estmato of Heght (Y n feet Estmated Vaance Regesson Estmato Estmated Vaance SRS D Although Table 3 confms ou smulaton esults. It should be emphased that the eample s used as an llustaton of the applcablt of ou poposed estmatos. 6. Conclusons In concluson D egesson estmato s to be used to mpove the populaton mean estmaton wheneve D s possble to be conducted. Refeences AL-SALEH M.F. and AL-KADIRI M.A Double anked set samplng. Statstcs and pobablt lettes 48(: 05-. DELL T.R. and CLUTTER J.L. 97. Ranked set samplng theo wth ode statstcs backgound. Bometcs 8: KAUR A. PATIL G.P. SINHA A.K. and TAILLIE C Ranked set samplng: an annotated bblogaph. Envonmental and Ecologcal Statstcs : MCINTYRE G.A. 95. A method fo unbased selectve samplng usng anked set. Austalan Jounal of Agcultual Reseach 3: PATIL G.P. SINHA A.K. and TAILLIE C Ranked set samplng: A bblogaph. Envon. Ecolog. Statst. 6: PLATT W. J. EVANS G.W. and RATHBUN S.L The populaton dnamcs of a longlved confe. The Ame. Natualst 3: SAMAWI H.M. 00. On double eteme anked set sample wth applcaton to egesson estmato. Meton L n. -: SAMAWI H.M. AHMED M.S. and ABU DAYYEH W Estmatng the populaton mean usng eteme anked set samplng. Bometcal Jounal 38 (5: SAMAWI H.M. and AL-SAGEER O.A. 00. On the estmaton of the dstbuton functon usng eteme and medan anked set samplng. Bometcal Jounal 43 (3: SAMAWI H.M. and MUTTLAK H.A Estmaton of ato usng anked set samplng. Bometcal Jounal 38 (6: SUKHATME P.V. and SUKHATME B.V Samplng theo of suves wth applcatons. Ames: Iowa state unvest Pess. TAKAHASI K. and WAKIMOTO K On unbased estmates of the populaton mean based on the statfed samplng b means of odeng. Ann. Inst. Statst. Math. 0: -3. YU L.H. and LAM K Regesson estmato n anked set samplng. Bometcs 53: Append Poposton. Unde (4. E ( β D β. 39

10 HANI M. SAMAWI and EMAN M. TAWALBEH Poof : Let ( ( / ( k k k C W W W W then ( k [] [] E CkE( Y[]. ( k E( ˆ β D E E CkY E CkE Y Note that snce E Y whee β 0 β. ( [] 0 ( β + β W and C 0 then k k ( ˆ D w k ( 0 + k ( k w( 0 k + k k ( E β E C β β W Poposton. Va ( β ( ( D e W W k E β C β CW β ˆ /. Poof: ( W W Y k [ ] k ( W ( W k Va ˆ ( βd Va k ( k Va Ck Y C Va Y k [ ] k [ ]. Usng Y [] β 0 + β Wk ( +ε and snce then Va ( [] β 0 β ( E Y + W Va Y [] e ( βd Net we pove Theoem 4. Poof of Theoem 4.: ( W( W k e e ( k ( k ( ˆ. k W W ( W W k ( E( YRe gd E E ( YRe gd ˆ D + βd ( E Y E Y W E ˆ Y + k D W m m [ ] β 30

11 REGRESSION ESTIMATOR USING DOUBLE RANKED SET SAMPLING Usng (4. we have Theefoe ( β β ( Re ( ( k m E Y E + ˆ W + ˆ W gd D D m k ( W ( R + β + β SS W. Hence Y s an unbased estmato of. ( gd β ( EY E Re +. ( Va ( Y E Va ( Y + Va E ( YRe gd ˆ Re β Va E Y gd Va E YD + D W. Fom ( we have E( Y β ( and then Also but Then + ( ( Re + β ( Va E Y Va gd Va β β Va β 3. m ˆ ( β ˆ Re β E Va Y gd E Va Y D + D W E Va Y D + W Va D ( ˆ D βd ( + Cov Y W ( ˆ [] D D k k k [] ( Cov Y β W Cov Y CY W m k k ( W CkVa( Yk [] m ( W e Ck 0. m ( Re ˆ ( β E V a Y gd E V a Y D + E W V a D m e E e + E ( W m ( m k e ( Wk [ ] W k 3

12 HANI M. SAMAWI and EMAN M. TAWALBEH cleal ths mples that whee theefoe Va Y (( ( W ( e E e + m ( ( Wk [] ( W ( ( ρ + E m ( m Z( Z k ( ρ + E m Z * ( k ( ( ρ + E m m S ( k W ( W ( k W ZW S m ( ( S Z k m k ( Re gd ρ + E + β m S m (. Receved 8 Novembe 00 Accepted 3 Octobe 00 3