Estimatig te Populatio Mea usig Stratified Double Raked Set Sample Mamoud Syam * Kamarulzama Ibraim Amer Ibraim Al-Omari Qatar Uiversity Foudatio Program Departmet of Mat ad Computer P.O.Box (7) Doa State of Qatar. E-mail: M.syam@qu.edu.qa Scool of Matematical Scieces Uiversity Kebagsaa Malaysia 4600 UKM Bagi Selagor Malaysia. E-mail: kamarulz@ukm.my Al al-bayt Uiversity Faculty of Scieces Departmet of Matematics P.O.Box 0040 Mafraq 5 Jorda. E-mail: alomari_amer@yaoo.com ICM 0-4 Marc Al Ai ABSTRACT: Stratified double raked set samplig () metod is suggested for estimatig te populatio mea. Te is compared wit te simple radom samplig (SRS) stratified simple radom samplig (SSRS) ad stratified raked set samplig (SRSS). It is sow tat estimator is a ubiased of te populatio mea ad more efficiet ta SRS SSRS ad SRSS. Also by we ca icrease te efficiecy of mea estimator for specific value of te sample size. KE ORDS: Simple radom samplig; raked set samplig; double raked set samplig; efficiecy.. ITRODUCTIO: Te raked set samplig metod wic was proposed by McItyre (95) to estimate mea pasture yields was developed ad modified by may autors to estimate te mea of te populatio. Dell ad Clutter (97) sowed tat te mea of te RSS is a ubiased estimator of te populatio mea watever or ot tere are errors i rakig. Al-Sale ad Al-Omari (00) suggested multistage raked set samplig (MSRSS) tat icrease te efficiecy of estimatig te populatio mea for specific value of te sample size. Jemai ad Al-Omari (006) suggested double quartile raked set samplig (DQRSS) for estimatig te populatio mea ad sowed tat te DQRSS mea is a ubiased estimator ad more efficiet ta te SRS RSS ad te QRSS if te uderlyig distributio is symmetric. Jemai ad Al-Omari (007) suggested multistage quartile raked set samplig (MQRSS) to estimate te populatio mea tey sowed tat te efficiecy of te mea estimator usig MQRSS ca be icreased for specific value of te sample size m by icrease te umber of stages. Also see Al-Omari ad Jaber (008) Bouza (00) Al-asser (007) ad Oyama et al. (008). I tis paper we suggest te stratified double raked set samplig () to estimate te populatio mea *Co-respodig Autor: M.syam@qu.edu.qa of symmetric ad asymmetric distributios. Te orgaizatio of tis paper is as follows: I Sectio we preset some of samplig metods. Estimatio of te populatio mea is give i Sectio. A simulatio study is cosidered i Sectio 4. Fially coclusios o te suggested estimator are itroduced i Sectio 5. SAMPIG METHODS:.. Stratified Simple Radom Samplig I stratified samplig te populatio of uits is first divided ito subpopulatios of uits respectively. Tese subpopulatios are o overlappig ad togeter tey comprise te wole populatio so tat. Te subpopulatios are called strata. To obtai te full beefit from stratificatio te values of te must be kow. e te strata ave bee determied a sample is draw from eac te drawigs beig made i differet strata. Te sample sizes witi te strata are deoted by respectively. If a simple radom sample is take i eac stratum te wole procedure is described as stratified simple radom samplig (SSRS)... Raked Set Samplig Te raked set samplig (RSS) suggested by McItyre (95) is coducted by selectig radom samples from te populatio of size uits eac ad rakig eac uit witi eac set wit respect to te variable of iterest. Te a actual measuremet is take of te uit wit te smallest rak from te first sample. From te secod sample a actual measuremet is take from te secod smallest rak ad te procedure is cotiued util te uit wit te largest rak is cose for actual measuremet from te -t sample. Tus we obtai a total of measured uits oe from eac ordered sample of size ad tis completed oe cycle. Te cycle may be repeated m times util m uits ave bee measured... Double Raked Set Samplig:
Mamoud I. Syam Kamarulzama Ibraim Amer I. Al-Omari Te DRSS procedure proposed by Al-Sale ad Al-Kadiri (000) depeds o selectig radom samples eac of size uits from te populatio ad rak eac sample wit respect to a variable of iterest. Te DRSS is described as follows: () Idetify elemets from te target populatio ad divide tese elemets radomly ito sets eac of size elemets. () Use te usual RSS procedure o eac set to obtai raked set samples of size eac. Apply te RSS procedure agai o step () to obtai a DRSS of size. Te cycle ca be repeated m times if eeded to get a sample of size m uits..4. Stratified Double Raked Set Samplig: I stratified samplig te populatio of uits is first divided ito subpopulatios of uits respectively. Tese subpopulatios are o overlappig ad togeter tey comprise te wole populatio so tat. Te subpopulatios are called strata. If te double raked set sample is used i eac stratum te wole procedure is described as stratified double raked set samplig (). To illustrate te metod let us take te followig example. Please otice tat te umber of subpopulatios (strata) is ot importat to be eve or odd. Example : ( ) Suppose we ave two strata ad i te first stratum we ave 7 elemets divided ito sets 9 elemets i eac set ad i te secod stratum we ave 64 elemets divided ito sets 6 elemets i eac set as te followig: Stratum (): Assume te 7 elemets are () () () () () () () () () () () After rakig te elemets i eac set we obtai () () () () () ad () () () () () () () () () () () () () () () () () () () () e will apply RSS o eac of te ie elemets to get tree sets as te followig: Set (): Set (): () () ( ) () () () ( ) () () () Set : ( ) () ow te elemets of te double raked set sample i te first () () stratum are. ( ) () Stratum (): Assume te 64 elemets are () () () () () 44 () () () () () () () () y y 44 y y () 44 44 After rakig te elemets i eac set we obtai () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () ad () () e will apply RSS o eac of te 6 elemets to get four sets as te followig: () () () () Set (): ( ) () Set (): Set : Set : () () () ( ) () ( ) () ( ) () () ow te elemets of te double raked set sample i te () () secod stratum are ( ) (). Terefore te uits are () () ( ) () () () ( ) (). ESTIMATIO OF THE POPUATIO MEA Te simple radom sample estimator of te populatio mea from a sample of size is give by SRS i it variace ( SRS ) i. Te stratified simple radom sample estimator of te populatio mea from a sample of size if te populatio is divided ito strata is give by SSRS SRS i i wit variace SSRS.
Estimatig te Populatio Mea usig Stratified Double Raked Set Sample Te estimator of te populatio mea for a RSS of size is give by RSS i ii () ( RSS ) wit variace () i i were () i is te mea of () i te it order statistics of a sample of size. ow assume tat te variable of iterest as desity f(x) wit absolutely cotiuous distributio fuctio F(x) mea ad variace. et be i.i.d f(x). et us deote to te cose elemets i RSS by * * * ad to te cose elemets i DRSS by. Te estimator of te populatio mea for a DRSS of size wit oe cycle i te stratum is defied as DRSS i Te stratified double raked set samplig () estimator of te populatio mea is give by were total populatio size. Te variace of is give by ( i ) i DRSS i i () () is te stratum size ad is te i i ( ) i E E. i i i ( i i i ) were Property. is a ubiased estimator of te populatio mea wic meas E. Proof. e ave E i i E i i i i E( i i ) It was sow by Al-Sale ad Al-Kadiri (000) tat i i E SRS. Property. Ad SSRS Also SRSS Proof. Te variace of is ( ) i * everteless i i for eac stratum tis implies ( SRSS ) * i i ( SSRS ) ( SRS ) 4. SIMUATIO STUD I tis sectio a simulatio study is coducted to ivestigate te performace for estimatig te populatio mea. Symmetric ad asymmetric distributios ave bee cosidered for samples of sizes 9458.
Mamoud I. Syam Kamarulzama Ibraim Amer I. Al-Omari assumig tat te populatio is partitioed ito two or tree strata. I te first table we ave two strata; eac of tem as a eve umber of samples 8 ad 6 wit total sample size 4. I te secod table we ave two strata oe of tem as a eve umber of samples ad te oter as a odd umber of samples 4 ad 5 wit total sample size 9. I te tird table we ave two strata; eac of tem as a odd umber of samples 5 ad 7 wit total sample size. I te fourt table we ave tree strata; eac of tem as a eve umber of samples 4 6 ad 8 wit total sample size 8. I te fift table we ave tree strata; eac of tem as a odd umber of samples 5ad 7 wit total sample size 5. I te sixt table we ave two strata; eac of tem as a eve umber of samples 0 ad 8 wit total sample size 8. Te simulatio was performed for te SRSS SSRS ad SRS data sets from differet distributios symmetric ad asymmetric. Te symmetric distributios are uiform ad ormal ad te asymmetric distributios are expoetial gamma ad weibull. Usig 00000 replicatios estimates of te meas variaces ad mea square errors were computed. Te efficiecy of relative to SSRS is defied by eff SSRS SSRS ad te efficiecy of relative to SRS is defied as eff SRS SRS. (5) ad te efficiecy of relative to SRSS is defied as eff SRSS SRSS (6) Te values of te relative efficiecy foud uder differet distributioal assumptios are provided i Tables 4 5 ad 6. Table. Te efficiecy of relative to SRSS SSRS ad SRS for 4 ad samples sizes 8 ad 6 SRSS SSRS SRS Uiform 6.67 4.56 4.77 (0) ormal (0).794 4.67.956 Expoetial () 5.97 8.764 8.5547 () 8.98 8.546 8.4 eibull () 7.469 7.467 7.500 Table. Te efficiecy of relative to SRSS SSRS ad SRS for 9 ad samples sizes 4 ad 5 Uiform (0) ormal (0) Expoetial () () eibull () SRSS SSRS SRS 9.67 4.9465 4.465 7.557 0.667 9.56 5. 5.66 4.6 4.07.990.89.974.7885.6544 Table. Te efficiecy of relative to SRSS SSRS ad SRS for ad samples sizes 5 ad 7 SRSS SSRS SRS Uiform 0.78.4675.07 (0) ormal 6.794 8.794 7.98 (0) Expoetial 6.89 8.54 7.040 () 6.9807 7.058 7.058 () eibull () 7.976 7.00 6.9 Table 4. Te efficiecy of relative to SRSS SSRS ad SRS for 8 ad samples sizes 4 6 ad 8 Uiform (0) ormal (0) Expoetial () (). SRSS SSRS SRS 59.96 6.458 60.746 45.896 48.947 47.87 0.879.945.879 0.7 9.0 8.90
Estimatig te Populatio Mea usig Stratified Double Raked Set Sample eibull () 9.68 9.5487 9.876 Table 5. Te efficiecy of relative to SRSS SSRS ad SRS for 5 ad samples sizes 5ad 7 Uiform (0) ormal (0) Expoetial () () eibull (). SRSS SSRS SRS 45.56 49.00 47.869 0.967.576.987 9.689 0.448 8.699 7.974 9.749 8.4 8.995 9.4457 8.875 Table 6. Te efficiecy of relative to SRSS SSRS ad SRS for 8 ad samples sizes 8 ad 0 Uiform (0) ormal (0) Expoetial () () eibull (). SRSS SSRS SRS 55.67 6.46 59.5000 4.5476 46.8769 45.4 9.5748 8.749 0.0990 8.589 7.846 9.9 0.8546 0.744 9.6667 () 0.465 0.79 0.40 0.40 eibull () 0.40 0.0594 0.067 0.0654 e 8ad strata 4 6 ad 8 Expoetial () 0.0485 0.05 0.056 0.055 () 0.4 0.7 0.5 0. eibull () 0.057 0.044 0.044 0.047 e 5 ad strata 5 ad 7. Expoetial () 0.05 0.066 0.078 0.0665 () 0.089 0.0908 0.868 0.66 eibull () 0.007 0.060 0.0565 0.054 e 8 ad two strata 0 ad 8 Expoetial () 0.047 0.05 0.056 0.055 () 0.5 0.7 0.5 0. eibull () 0.087 0.044 0.044 0.047 From te tables we ca otice tat greater efficiecy is attaied usig metod as opposed to te oter cotedig metods tat ave bee discussed we estimatig te populatio mea of te variable of iterest. e te performace of are compared to eiter SRSS SSRS or SRS it is foud tat is more efficiet as sow by all te values of relative efficiecy wic are greater ta. e te performaces of te suggested estimators are compared te efficiecy of te suggested estimator is foud to be more superior we te uderlyig distributios are symmetric as compared to asymmetric. Te relative efficiecy of estimator to tose estimators based o SRS SSRS ad SRSS are icreasig as te sample size icreases. 5. COCUSIOS Table 7. Te values of bias of SRSS SSRS ad SRS for differet distributios ad differet umbers of strata e 4 ad two strata 8 ad 6 SRSS SSRS SRS Expoetial() 0.084 0.09 0.076 0.07 () 0.99 0.7 0.90 0.856 eibull () 0.09 0.0475 0.057 0.056 e 9 ad two strata 4 ad 5 Expoetial () 0.0766 0.094 0.5 0.560 () 0.5 0.798 0.4697 0.497 eibull () 0.878 0.0946 0.00 0.04 e ad two strata 5 ad 7 Expoetial () 0.078 0.044 0.085 0.08 I tis paper we ave suggested a ew estimator of te populatio mea usig. Te performace of te estimator based o is compared wit tose foud usig SRSS SSRS ad SRS for te same umber of measured uits. It is foud tat produces estimator of te populatio mea tat is ubiased ad is more efficiet ta SRSS SSRS ad SRS. Tus sould be more preferred ta SRSS SSRS ad SRS for bot symmetric ad asymmetric distributios. REFERECES. Al-Sale M.F. & Al-Hadrami S. A. (00). Estimatio of te mea of expoetial distributio usig movig extremes raked set samplig. Statistical Papers 44: 67-8.. Al-Sale M.F. ad Al-Kadiri M. (000). Double raked set samplig. Statistics Probability etters 48 05 -.
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