New atio Estimators Usig Correlatio Coefficiet Cem Kadilar ad Hula Cigi Hacettepe Uiversit, Departmet of tatistics, Betepe, 06800, Akara, Turke. e-mails : kadilar@hacettepe.edu.tr ; hcigi@hacettepe.edu.tr Abstract : We opose a class of ratio estimators for the estimatio of populatio mea b adaptig the estimators i igh ad Tailor (003) to the estimators i Kadilar ad Cigi (004). We obtai mea square error (ME) equatios for all oposed estimators, ad fid theoretical coditios that make each oposed estimator more efficiet tha the traditioal estimators, ad similarl for compariso to the ratio estimator i igh ad Tailor (003), ad for those i Kadilar ad Cigi (004). I additio, these coditios are supported b a umerical eample. Ke words : atio estimator, auiliar variable, simple radom samplig, efficiec. 000 AM Classificatio : 6 D 05, 6 G 05
1. Itroductio The classical ratio estimator for the populatio mea of the stud variable defied b r X (1) is where ad are the sample meas of stud ad auiliar variables, respectivel, ad it is assumed that the populatio mea X of the auiliar variable is kow. The ME of this estimator is as follows: where ad f ME ( r ) ( + [ θ]) () f respectivel; N ; is the sample size; N is the umber of uits i the populatio; are the populatio variaces of auiliar ad stud variables, ad X C θ. Here C ad C are the populatio C coefficiets of variatio of auiliar ad stud variables, respectivel (Cochra, 1977, pages 151-154). igh ad Tailor (003) suggested the followig ratio estimator: T ( X + ) (3) + where is the correlatio coefficiet betwee auiliar ad stud variables. The ME of this ratio estimator is as follows: f ME ( T ) ( + ω[ ω θ]) (4)
X where ω. X + where ad Kadilar ad Cigi (004) suggested the followig ratio estimators: KC ( X ) X + b 1 (5) + b( X ) [ X + β ( )] (6) + β ( ) KC ( X C + b( X ) KC3 + ) (7) + C ( X ) [ Xβ ( C + b KC 4 ) + ] (8) β ( ) + C ( X ) [ X C + β ( ) + b ] (9) C + β ( ) KC 5 β ( ) is the populatio coefficiet of the kurtosis of the auiliar variable s b is the regressio coefficiet. Here s is the sample variace of s auiliar variable ad s is the sample covariace betwee the stud ad auiliar variables. Kadilar ad Cigi (004) obtaied the ME equatio of these ratio estimators as follows: ME f ( ) [ + ( )] ; i 1,,..., 5 KC i KC i (10) 3
where KC X 1 ; KC ; X + β ( ) KC3 ; X + C KC 4 Xβ β ( ) ( ) + C ad KC5 C. XC + β ( ). uggested Estimators Adaptig the estimator give i (3) to the estimators give i (5)-(9), we develop ew ratio estimators usig the correlatio coefficiet as follows: ( X ) ( + + b 1 X ) (11) + + b( X ) ( XC + ) (1) C + ( X C + b( X ) + ) (13) + C ( X ) [ X β ( + + b ) 4 β ( ) + ] (14) ( X ) [ X + β ( ) + b ] (15) + β ( ) 5 We obtai the ME equatio for these oposed estimators as ME f ( ) [ + ( )] ; i 1,,..., 5 i i (16) 4
where 1 ; X + C ; XC + ; X + C ( ) ( ) + β 4 ad Xβ 5. (for details see Appedi) X + β ( ) 3. Efficiec Comparisos I this sectio, we tr to obtai the efficiec coditios for the oposed estimators b comparig the ME of the oposed estimators with the ME of the sample mea, traditioal ratio estimator ad the ratio estimators suggested b igh ad Tailor (003) ad Kadilar ad Cigi (004). It is well kow that uder simple radom samplig without replacemet (WO) the variace of the sample mea is V f. (17) ( ) We first compare the ME of the oposed estimators, give i (16), with the variace of the sample mea. We have the followig coditio: Let υ ( ) < V ( ) ; i 1,,..., 5 ME i. i < 0, > υ i (18) Whe this coditio is satisfied, oposed estimators are more efficiet tha the sample mea. 5
ecodl, we compare the ME of the oposed estimators with the ME of the classical ratio estimator, give i (). We have the followig coditio: ME i ( ) < ME( ) ; i 1,,..., 5 > υ i r. < ( + θ) θ, i. (19) Whe this coditio is satisfied, oposed estimators are more efficiet tha the traditioal ratio estimator. Thirdl, we compare the ME of the oposed estimators with the ME of the estimator i igh ad Tailor (003), give i (4). We have the followig coditio: ME i ( ) < ME( ) ; i 1,,..., 5 > υ i T < ω ( ω + ωθ) ωθ, i. (0) Whe this coditio is satisfied, oposed estimators are more efficiet tha the ratio estimator, suggested b igh ad Tailor (003). Fiall, we compare the ME of the oposed estimators with the ME of the estimators i Kadilar ad Cigi (004), give i (10). We have the followig coditio: ME ( ) < ME( ) ; i 1,,..., 5 i KC j ad j 1,,,5. < i KC j (1) Whe this coditio is satisfied, oposed estimators are more efficiet tha the ratio estimators, suggested b Kadilar ad Cigi (004). We ca eamie the 6
coditio (1) for each oposed estimator. For eample, whe we take i j 1, we obtai the coditio: X < X + X + ( + ) > 0 X As is positive i ratio estimatio we have the followig coditio: > X This coditio is alwas satisfied if the auiliar variable has positive data. I other words, first oposed estimator is more efficiet tha first ratio estimator i Kadilar ad Cigi (004) whe the data are positive. Detail comparisos for the other oposed estimators ca also be studied i the same wa. Note that the efficiec comparisos amog the oposed estimators also result the similar coditio with (1) as follows: i < ; i j 1,,...,5 j () Whe this coditio is satisfied, ith oposed estimator is more efficiet tha jth oposed estimator. 4. Numerical Eample I this sectio, we appl the traditioal ratio estimator, give i (1), the igh- Tailor ratio estimator, give i (3), Kadilar-Cigi ratio estimators, give i (5)-(9) ad oposed estimators, give i (11)-(15), to data whose statistics are give i Table 1. We assume to take the sample size 50 from N00 usig WO. The ME of these estimators are computed as give i (), (4), (10) ad (16) ad these estimators are compared to each other with respect to their ME values. 7
Table 1 Data tatistics N 00 500 KC 1 0.00 1 19.31 50 X 5 KC 6.67 19.65 0.90 β ( ) 50 KC 3 18.5 18.37 C 15 θ 6.75 KC 4 19.97 4 19.99 C ω 0.97 KC 5 10.00 5 6.00 From Table, we uderstad that the most efficiet estimator is fifth oposed estimator. Whe we eamie the coditios, determied i ectio 3, for this data set, we see that all of them are satisfied for fifth oposed estimator as follows: 0.81 ad υ 5 0. 0166 the coditio (18) is satisfied. 0.81 ad υ( + θ) 0. 388 5 the coditio (19) is satisfied. 0.81 ad υ( ω + ωθ) 0. 317 5 the coditio (0) is satisfied. 5 < ; i 1,,...,5 KC i the coditio (1) is satisfied. 5 < ; j 1,,3,4 j the coditio () is satisfied. 8
Therefore, we suggest that we should appl fifth oposed estimator to this data set. It is worth poit out that the traditioal ratio estimator is more efficiet tha the ratio estimator, suggested b igh ad Tailor (003), for this data set. Table ME Values of atio Estimators Estimators ME Estimators ME Estimators ME 843750.00 KC1 17531.50 1 17488.14 r 65650.00 KC T 666.3 KC3 161979.17 174786.74 17317.58 17963.48 KC 4 17564.61 4 KC5 17590.9 16406.50 5 161757.1 5. Coclusio We develop some ratio estimators usig the correlatio coefficiet ad theoreticall show that the oposed estimators have a smaller ME tha the traditioal, the igh ad Tailor's (003) ad Kadilar ad Cigi's (004) ratio estimators i certai coditios. These theoretical coditios are also satisfied b the results of a umerical eample. I future work, we hope to adapt the ratio estimators, eseted here, to ratio estimators i stratified radom samplig as i Kadilar ad Cigi (003; 005). Appedi To the first degree of apoimatio, the ME of the third oposed estimator ca be foud usig the Talor series method defied b 9
( ) δσ δ ME (A.1) where ( c, d ) h ( c d ) h, δ, c B,, X d B,, X Σ f (see Wolter, 003, pages 1-8). Here B ; h ( c d ) h (, ) 3, i (13) ad deotes the populatio covariace betwee stud ad auiliar variables. Accordig to this defiitio, we obtai δ for the third oposed estimator as [ 1 B] δ. 3 We obtai the ME equatio of the third oposed estimator usig (A.1) as follows: ME f ( ) ( B + + B + B ) f f f f ( + + ) ( + ) [ + ( )]. + + + 10
We would like to remark that the ME equatios of the other oposed estimators ca easil be obtaied i the same wa. efereces Cigi, H. (1994). amplig Theor. Hacettepe Uiversit Press. Cochra, W.G. (1977). amplig Techiques. Joh Wile ad os, New-ork. Kadilar, C. ad Cigi, H. (005). A New atio Estimator i tratified adom amplig. Commuicatios i tatistics: Theor ad Methods\QT{it}{,} 34, 597-60. Kadilar, C. ad Cigi, H. (004). atio Estimators i imple adom amplig. Applied Mathematics ad Computatio, 151, 893-90. Kadilar, C. ad Cigi, H. (003). atio Estimators i tratified adom amplig. Biometrical Joural, 45, 18-5. igh, H. P. ad Tailor,. (003). Use of Kow Correlatio Coefficiet i Estimatig the Fiite Populatio Mea. tatistics i Trasitio, 6, 555-560. Upadhaa, L. N. ad igh, H. P. (1999). Use of Trasformed Auiliar Variable i Estimatig the Fiite Populatio Mea. Biometrical Joural, 41, 67-636. Wolter, K. M. (003). Itroductio to Variace Estimatio. iger-verlag. 11