Ratio Estimators Using Coefficient of Variation and Coefficient of Correlation

Transcription

1 Moder Applied iee; Vol. 8, No. 5; 04 IN E-IN Publised b Caadia Ceter of iee ad Eduatio atio Estimators Usig Coeffiiet of Variatio ad Coeffiiet of Correlatio Praad aggam Departmet of tatistis, Fault of iee, ilpakor Uiversit, Nakor Patom, Tailad Correspodee: Praad aggam, Departmet of tatistis, Fault of iee, ilpakor Uiversit, Nakor Patom 73000, Tailad. praad@su.a.t eeived: Jue 4, 04 Aepted: Jue 0, 04 Olie Publised: August 6, 04 doi:0.5539/mas.v85p70 U: ttp://d.doi.org/0.5539/mas.v85p70 Abstrat Tis paper itrodues ratio estimators of te populatio mea usig te oeffiiet of variatio of stud variable ad auiliar variables togeter wit te oeffiiet of orrelatio betwee te stud ad auiliar variables uder simple radom samplig ad stratified radom samplig. Tese ratio estimators are almost ubiased. Te mea square errors of te estimators ad teir estimators are give. ample size estimatio i bot samplig desigs are preseted. A optimal sample size alloatio i stratified radom samplig is also suggested. Based o teoretial stud, it a be sow tat tese ratio estimators ave smaller ME ta te ubiased estimators. Moreover, te empirial stud idiates tat tese ratio estimators ave smallest ME ompared to te eistig oes. Kewords: ratio estimator, sample size estimatio, oeffiiet of variatio, oeffiiet of orrelatio. Itrodutio Cosider a populatio of N uits wit observatios, i i for i,,, N were i is a value of stud variable ad i is a value of auiliar variable. Uder simple radom samplig witout replaemet, a ubiased estimator of te populatio mea N i is te sample mea i. Te variae of te N i i ubiased estimator is f V, (.) N were f ad i. N N i X A ommo ratio estimator is were X ad are te populatio ad sample meas of te auiliar variable, respetivel. Te effiie of te ratio estimator depeds o te oeffiiet of variatio of auiliar variable C ad oeffiiet of variatio of stud variable C. Murt (964) as suggested tat C if, te ratio estimator performs better ta te ubiased estimator were is te orrelatio C oeffiiet betwee ad. Te approimate bias ad mea square error (ME) of te ratio estimator are as follows: f B X X, (.) f ME, (.3) were, X N i X ad N N i i X. We te C is kow, N i i 70

2 Moder Applied iee Vol. 8, No. 5; 04 issodia ad Dwivedi (98) as proposed a modified ratio estimator, X C D C. Te approimate bias ad ME of te estimator are f B D X X, (.4) f ME D, (.5) were X C. ampat (005) used te oeffiiet of variatio of te stud variable to improve te f ubiased estimator as C. Te approimate bias ad ME of tis estimator are f B C, (.6) f f ME C. (.7) I additio, tere are several autors, su as Upadaa ad ig (999), ig ad Tailor (003), wo ave developed various ratio estimators uder simple radom samplig. If te stud variable as differet mea values i differet subpopulatios, it is advatageous to draw a sample b stratified radom samplig. I stratified samplig, a populatio is partitioed ito strata. A stratum otais N uits wit observatios i, i were,,, ad i,,,n. A ubiased estimator of uder stratified radom samplig is give b N st W were W is te stratum N weigt ad is te sample mea of te stud variable i stratum. Te variae of te ubiased estimator is were st V W, (.8) f, f N is samplig fratio i stratum, is a sample size i stratum ad is te variae of te stud variable i stratum. Tere are two tpes of ratio estimators i stratified radom samplig, amel ombied ad separate ratio estimators. Te ombied ratio estimator is give b st C X, were st W is a ubiased estimator of te st populatio mea X ad is te sample mea of auiliar variable i stratum (Cora, 977). Te approimate mea squared error of te ombied ratio estimator is C ME W, (.9) were is te populatio ratio, is te variae of auiliar variable i stratum ad is te X ovariae betwee auiliar ad stud variables i stratum. Te approimate bias of te ombied ratio estimator is B C W. (.0) X X Te separate ratio estimator is give b estimator a be give b X W. Te approimate ME of te separate ratio ME W, (.) 7

3 Moder Applied iee Vol. 8, No. 5; 04 were X is te populatio ratio i stratum. Te approimate bias of te separate ratio estimator is B W. (.) X X Kadilar ad Cigi (003) ave proposed several ombied ratio estimators. Te simplest oe based o te issodia ad Dwivedi (98) estimator is defied as KC st W X C W C were C is te oeffiiet of variatio of te auiliar variable i stratum. Te ME ad bias of tis estimator are approimated as follows: were KC X st KC KC KC KC ME W, (.3) B W st. W X C KC KC XKC XKC, (.4) Kadilar ad Cigi (005) ave improved te ombied ratio estimator i stratified radom samplig based o te estimator itrodued b Prasad (989). However, tese estimators deped o several ukow parameters ad terefore arediffiult to use. I etios ad 3, te ratio estimators based o te oeffiiet of variatio ad orrelatio i simple radom samplig ad stratified radom samplig are itrodued, respetivel. Te approimate bias ad ME of te estimator are derived. A estimator of te ME is give. Te sample size estimatio ad a optimal alloatio of sample size i stratified radom samplig is preseted. Te ompariso of te effiie betwee te proposed estimator ad ubiased estimator is teoretiall provided. Hpotetial populatios are used to ompare te properties of te preseted estimators wit te eistig oes.. Estimatio i imple adom amplig. Parameter Estimatio Cosider te followig ratio estimator for te populatio mea of te stud variable, X (.) were is a real ostat to be determied su tat te ME is miimized. Note tat we is equal to 0, tis estimator is redued to te usual ratio estimator ad we is equal to C tis estimator beome te estimator of issodia ad Dwivedi (98). To obtai te ME ad bias of te estimator (.), let e ad Ee e X X V X. It a be sow tat. Te estimator E e 0, approimatio, we obtai E e 0, Ee, Eee V a be writte as Cov, X ad e e. Usig Talor series e e e e e.... We te terms of degree greater ta two are igored, we get te approimate bias of te estimator as f B, (.) X X 7

4 Moder Applied iee Vol. 8, No. 5; 04 were. imilarl, Talor s formula a be used to approimate ME of te estimator as X f ME. (.3) * Miimizig (.3) wit respet to, we get te optimum value of as X. ubstitutig * for i (.), (.) ad (.3) ad usig algebra, we obtai te optimum estimator, its bias ad ME as follows, XC * C X XC, (.4) B * 0, (.5) f ME *. (.6) Note tat te optimum estimator is almost ubiased ad its ME is alwas smaller ta te variae of te ubiased estimator. I additio, te optimum estimator a be applied for bot populatios wit positive ad egative oeffiiet of orrelatio. For a sample estimate of te ME, oe a substitute te sample estimate of wi gives f ME ˆ * s, (.7) were s is te sample variae of te stud variable.. ample ize Estimatio ample size estimatio is oe of te importat aspets i sample surves. If te sample size is too small, te samplig error ma be too large. However, too large sample size implies a waste of resoures. We would like to speif a sample size tat is suffiietl large to esure a ig probabilit tat te estimate loses to te parameter. Uder simple radom samplig, te populatio mea of te stud variable is estimated wit te optimum estimator *. To obtai te desired sample size, oe a speif te margi of error d ad te probabilit su tat P * d 73. Uder some teial oditios as sow i ott ad Wu (98) ad Hajek (960), we a sow tat * is asmptotiall ormal distributed wit mea ad variae ME *. To obtai te absolute preisio, we a fid a value of tat satisfies d/ ME * z / were z / deotes te upper / poit of te stadard ormal distributio. olvig for, we ave 0, (.8) 0 N z were 0. If te populatio size N is large relative to te sample size, te formula of te d sample size redues to 0..3 Compariso of Effiie I tis setio, te properties of te estimators i simple radom samplig are ompared. Te relative effiie of te optimum estimator ad ubiased estimator is osidered as follows: e, * E E *. (.9) Tis sows tat te optimum estimator is alwas more effiiet ta te ubiased estimator beause 0. Te effiie depeds o te oeffiiet of orrelatio:if te oeffiiet of orrelatio ireases te te effiie also ireases.

5 Moder Applied iee Vol. 8, No. 5; 04 To ompare te properties of te optimum estimator wit te oters, we osider potetial populatios wit var arateristis. I tis work, te oeffiiets of orrelatio i te populatios are 0., 0.,, 0.9. I ea populatio, te oeffiiets of variatios are C 0., C 0. ad te populatio meas X 5000,, 5000,. Wit varig sample sizes, te biases ad MEs of te estimators are give i Table ad Table, respetivel. Te biases ad MEs are omputed b te formulas i te previous setios. I Table, as epeted, te absolute bias of te ubiased ad optimum ratio estimators are alwas equal to 0. Te estimator as te largest absolute bias amog te ompared estimators. Te bias of te estimator is egative beause te estimator is ostruted b usig a ostat i wi its value less ta multiplig te ubiased estimator. Te bias of te estimator does ot deped o te oeffiiet of orrelatio. Observe tat te absolute biases of te estimator D are smaller ta of te estimator. Give a sample size, we te oeffiiet of orrelatio ireases te absolute bias of te two estimators ad D derease. Give a oeffiiet of orrelatio, te absolute bias of, ad D derease we te sample size ireases. Table sows tat te optimum ratio estimator as smallest ME amog te ompared estimators. Te MEs of te two estimators ad do ot deped o te oeffiiet of orrelatio. We 05. te MEs of te two estimators ad D are less ta tose of te ubiased estimator. Give a sample size, we te oeffiiet of orrelatio ireases te MEs of te tree estimators, D ad * derease. Give a oeffiiet of orrelatio, te MEs of all estimators derease we te sample size ireases. Table. Biases of te estimators i simple radom samplig B B B B B D * 74

6 Moder Applied iee Vol. 8, No. 5; 04 Table. MEs of te estimators i simple radom samplig ME ME ME ME ME D * 3. Estimatio i tratified adom amplig 3. Parameter Estimatio I stratified radom samplig, we estimator a be modified as X, C, C ad i stratum are kow, te separate ratio X C W. (3.) C X X C _C ie tis estimator is ostruted from te optimum ratio estimator, we all tis estimator optimum separate ratio estimator. To obtai te ME ad bias of te optimum separate ratio estimator, applig te ME ad bias X C of * uder simple radom samplig to draw i stratum,ields C B 0, (3.) _C _C ME W, (3.3) For estimatig te ME _C, we substitute te sample estimates to obtai 75

7 Moder Applied iee Vol. 8, No. 5; 04 _C ME ˆ W s. (3.4) Note tat te bias of te optimum separate ratio estimator is te umulative bias of a optimum ratio estimate i ea stratum wi loses to zero. I additio, we foud tat te ME of te estimator is also smaller ta te variae of te ubiased estimator i (.8). 3. Optimum ample ize Alloatio Give a total sample size ad usig te optimum separate ratio estimator, oe ma oose ow to alloate te sample size amog te strata. I tis setio, te alloatio seme wi miimizes te ME of te estimator b fiig te total sample size is osidered. Tat is, we eed te values of,,, wi miimize ME W subjet to te oditio. Te sample size alloated to ea stratum is _C N ;,,,. (3.5) N Tus, te optimum seme alloates larger sample sizes to strata wit larger variaes ad larger stratum sizes but smaller sample sizes to strata wit larger oeffiiets of orrelatio. 3.3 ample ize Estimatio Te formula (3.5) gives i terms of, but i pratie, we do ot et kow wat value of is. Tis setio presets a formula for te determiatio of uder te optimum sample size alloatio. It is assumed tat te optimum separate ratio estimate as a speified mea squared error M. We, N ad N are all suffiiet large ad te teial oditios i ott ad Wu (98) old, we a sow tat te estimator as also te asmptoti ormal distributio. If te margi of error d as bee give, te _C / M d/z. et w, were olvig for, we ave z/ W 0 were d 3.4 Compariso of Effiie w W W. o, te mea square error of _C is W d M W w N z/. 0 z W Nd /., (3.6) I tis setio, we ompare te properties of te proposed optimum separate ratio estimator wit te eistig oes i stratified radom samplig. Te relative effiie of te optimum separate ratio estimator ad ubiased estimator is W st _ C W e,. (3.7) Tis sows tat te optimum separate ratio estimator is alwas more effiiet ta te ubiased estimator. Te effiie depeds o te oeffiiet of orrelatio i stratum. If te orrelatio oeffiiet ireases te te effiie also ireases. 76

8 Moder Applied iee Vol. 8, No. 5; 04 To ompare te properties of te optimum separate ratio estimator wit te oter estimators, we osider te followig potetial populatios. Ea populatio osists of N = 50,000 uits ad is divided ito = strata of wi sizes are N 0, 000 ad N 30, 000. Te oeffiiets of variatios are C C 0. ad C C 0.3. Te populatio meas are give b X 500 ad X, 000. Te oeffiiets of orrelatios are 0., 0.,, 0.9. We set te total sample size 50 wit tree alloatios, amel equal alloatio, proportioal alloatio N ad optimal alloatio as i (3.5). Te biases ad N MEs of te estimators are give i Table 3 ad Table 4, respetivel. Te biases ad MEs are omputed b te formulas i te above setios. I Table 3, te absolute biases of te ubiased ad optimum separate ratio estimators are alwas equal to 0. Te absolute biasof te estimator is smallest amog te ompared estimators. Te absolute bias of te estimator KC is smaller ta tat of te estimator C. Give a sample size alloatio, we te oeffiiet of orrelatio ireases te absolute biases of te tree estimators C,, ad KC derease. Usig te optimum alloatio, te absolute biases of te estimators C, ad KC are smallest amog te tree alloatios. Table 4 presets tat te optimum separate ratio estimator gives te smallest ME amog te ompared estimators. Observe tat te ME of te ubiased estimator st does ot deped o te oeffiiet of orrelatio beause it does ot use te iformatio of te auillar variable. We 05., te MEs of te estimators C, ad KC are less ta tat of te ubiased estimator. Give a sample size alloatio, we te oeffiiet of orrelatio ireases, te MEs of te estimators C,, KC ad _C derease. Te MEs of all estimators are smallest uder te optimum alloatio. Table 3. Biases of te estimators i stratified radom samplig Alloatio Bst BC B KC B equal proportio optimum B _C

9 Moder Applied iee Vol. 8, No. 5; is te oeffiiet of orrelatio i te wole populatio. Table 4. ME of te estimators i stratified radom samplig Alloatio ME st ME C ME ME KC ME _ C equal proportio optimum is te oeffiiet of orrelatio i te wole populatio. 4. Disussio I simple radom samplig, te optimum ratio estimator ad its variae estimate deped o te oeffiiet of variatio, te oeffiiet of orrelatio ad te mea of te auiliar variable i te wole populatio. imilarl, te optimum separate ratio estimator ad its variae estimate are i terms of te oeffiiets of variatio, te oeffiiet of orrelatio ad te meas of te auiliar variable i all strata. I pratie, sample estimates of tese parameters ma be used to substitute i te formulas of tese estimates. I simple radom samplig, te relative effiie of te optimum ratio estimator ad te ubiased estimator depeds o te oeffiiet of orrelatio. We te oeffiiet of orrelatio betwee te stud ad auillar 78

10 Moder Applied iee Vol. 8, No. 5; 04 variable is weak, te te relative effiie will be low so tat te ubiased estimator is almost as good as te ratio estimators. For eample, if 0., te e, * 04.. Tis meas tat if we irease te sample size about 4.%, te ubiased estimator will ave te most effiie amog te estimators i te lass of :. Terefore, i ase of 0. we suggest usig te ubiased estimator beause it X uses ol te iformatio of te stud variable ad we do ot eed to ollet te data of te auillar variables. I stratified radom samplig, we Ma 0., we also reommed te ubiased estimator. For future studies, we a osider applig te ratio estimator i adaptive samplig semes as suggested b Tompso (990) ad aggam (03) for. Akowledgemets Tis work was supported b te Fault of iee, ilpakor Uiversit, Nakor Patom, Tailad. eferees Cora, W. G. (977). amplig Teiques. Nework: Jo Wile ad os. Hajek, J. (960). imitig distributios i simple radom samplig from a fiite populatio. Publiatio of te Matematial Istitute of te Hugaria Aadem of ieees, 5, Kadilar, C., & Cigi, H. (003). atio estimators i stratified radom samplig. Biometrial Joural, 45(), 8 5. ttp://d.doi.org/0.00/bimj Kadilar, C., & Cigi, H. (005). A ew ratio estimator i stratified radom samplig. Commuiatio i tatistis Teor ad Metod, 34, ttp://d.doi.org/0.08/ta Murt, M. N. (964). Produt metod of estimatio. aka, 6, Prasad, B. (989). ome improved ratio tpe estimators of populatio mea ad ratio ifiite populatio sample surves. Commuiatio i tatistis Teor ad Metod, 8(), ttp://d.doi.org/0.080/ ampat. (005). amplig Teor ad Metods. Alpa iee Iteratioal td. Harrow, U.K. aggam, P. (03). Uequal Probabilit Iverse Adaptive Cluster amplig. Ciag Mai Joural of iee. 40(4), ott, A., & Wu, C. (98). O te asmptoti distributio of ratio ad regressio estimators.joural of Ameria tatistial Assoitatio, 76(373), ttp://d.doi.org/0.080/ ig, H. P., & Tailor,. (003). Use of kow orrelatio oeffiiet i estimatig te fiite populatio mea. tatistis i Trasitio, 6, isodia, B. V.., & Dwivedi, V. K. (98). A modified ratio estimator usig oeffiiet of variatio of auiliar variable. Joural of Idia oiet of Agriultural tatistis, 33, 3 8. Tompso,. K. (990). Adaptive luster samplig. Joural of te Ameria tatistial Assoiatio, 85, Upadaa,. N., & ig, H. P. (999). Use of trasformed auiliar variable i estimatig te fiite populatio mea. Biometrial Joural, 4(5), ttp://d.doi.org/0.00/(ici)5-4036(99909)4:5<67::aid-bimj67>3.0.co;-w Coprigts Coprigt for tis artile is retaied b te autor(s), wit first publiatio rigts grated to te joural. Tis is a ope-aess artile distributed uder te terms ad oditios of te Creative Commos Attributio liese (ttp://reativeommos.org/lieses/b/3.0/). 79