EFFICIENT CLASSES OF RATIO-TYPE ESTIMATORS OF POPULATION MEAN UNDER STRATIFIED MEDIAN RANKED SET SAMPLING

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1 Pak. J. Sais. 016 Vol. 3(6) EFFICIENT CASSES OF RATIO-TYPE ESTIMATORS OF POPUATION MEAN UNDER STRATIFIED MEDIAN RANKED SET SAMPING akkar Kan 1 Javid Sabbir and Ce Kadilar 3 1 Higer Eduacaion Deparen Kber Pakunkwa Pakisan. Eail: lakkarkan.sa@gail.co Deparenof Saisics Quaid-i-Aza Universi Islaabad Pakisan. Eail: javidsabbir@gail.co 3 Depareof Saisics Haceepe Universi Beepe Ankara Turke. Eail: kadilar@aceepe.edu.r Corresponding auor ABSTRACT In is paper we propose wo efficien classes of raio-pe esiaors for esiaing e finie populaion ean ( Y ) under sraified edian ranked se sapling MRSS ) using e known auiliar inforaion. Te biases and ean squared errors ( MSEs ) of e proposed classes of raio-pe esiaors are derived upo firs order of approiaion. Te proposed esiaors are copared wi soe copeior esiaors. I is deonsraed roug siulaion sud a e proposed raio-pe esiaors based on S MRSS are ore efficien an e corresponding esiaors in sraified ranked se sapling RSS ) given b Mandowara and Mea [13]. KEYWORDS Sraified ranked se sapling Mean squared error Raio-pe esiaors Sraified edian ranked se sapling. 000 Maeaics Subjec Clasificaion: 6D INTRODUCTION Ranked Se Sapling ( RSS ) ecnique was firs inroduced b Mclnre [14] and S RSS was suggesed b Saawi and Mulak [16] o obain ore efficien esiaor for populaion ean. Te also proposed an esiaor of populaion raio in RSS and sowed a i as less variance as copared o raio esiaor in siple rando sapling (SRS). Takaasi and Wakioo [0] sowed a e saple ean under RSS is an unbiased esiaor of e populaion ean and ore precise an e saple ean esiaor under SRS. Using S RSS e perforances of e cobined and separae raio esiaes was obained b Saawi and Sia [17]. Mandowara and Mea [13] ave adoped e Kadilar and Cingi [5] esiaors in S RSS and obained ore efficien raio-pe esiaors. Al-Sale and Al-Kaddiri [] ave inroduced e concep of double ranked se 0 Pakisan Journal of Saisics 475

2 476 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean sapling ( DRSS ) and sowed a DRSS esiaor is ore efficien an e usual RSS esiaor in esiaing e finie populaion ean. As a odificaion of e RSS Mulak [17] as suggesed e edian ranked se sapling ( MRSS ) eod for esiaing e populaion ean. Jeain and Al-Oari [4] ave suggesed ulisage edian ranked se sapling for esiaing e populaion ean. Al-Oari [1] as inroduced odified raio esiaors in MRSS. Kouncu [1] as proposed raio and eponenial pe esiaors in MRSS. Kan and Sabbir [78] proposed classes of Harel-Ross pe unbiased esiaors in RSS and S RSS. Kan e al. [11] proposed unbiased raio esiaors of e finie populaion ean in [910] also proposed efficien classes of esiaors in RSS. S RSS. Kan and Sabbir In is paper we use e idea of SMRSS in esiaing e finie populaion ean and coparison is ade wi Mandowara and Mea [13] esiaors.. STRATIFIED RANKED SET SAMPING In S RSS for e srau firs coose independen rando saples eac of size ( =1... ). Ranked e observaions in eac saple and use RSS procedure o ge independen ranked se saples eac of size o ge 1... = observaions. Tis coplees one ccle of S RSS. Te wole process is repeaed r ies o ge e desired saple size n = r. To obain Bias and MSE of e esiaors we define: = Y (1 ) = X (1 ) RSS ] 0 RSS ) 1 suc a E( i ) = 0 ( i = 01) were and W and W 1 0 =1 i=1 [ i: ] r E( ) = C W W 1 1 =1 i=1 ( i: ) r E( ) = C W W 0 1 =1 i=1 [ i: ] ( i: ) r E( ) = ( C C W W = = ( : ) = ( : ) X Y X [ i: ] ( i: ) [ i: ] W ( i: ) = Y. [ i: ] ([ i: ] i i

3 Kan Sabbir and Kadilar 477 Here = C and = C Y X Y and X are e populaion eans; = E and ( i: ) = E ( i: ) ; [ i: ] [ i: ] Y and srau of e variables Y and X respecivel; coefficien beween eir respecive subscrips in e X are e populaion eans of e is e populaion correlaion srau. Using S RSS e cobined raio esiaor of populaion ean ( Y ) given b Saawi and Sia [17] is defined as X = RSS ) R RSS )1 RSS ] (1) were RSS ] W [ RSS ] and RSS ) W ( RSS ) are e unbiased esiaors = =1 = =1 N of populaion eans Y and X respecivel; W = N is e known srau weig N is e srau size N is e oal populaion size and is e oal nuber of sraa ( =1... ). Following Sisodia and Dwivedi [19] Mandowara and Mea (014) ave suggesed a odified raio-pe esiaor for populaion ean ( Y ) using S RSS wen populaion coefficien of variaion of e auiliar variable for e W X C =. =1 R RSS ) RSS ] W ( RSS ) C =1 srau C is known as Following Kadilar and Cingi [6] Mandowara and Mea (014) ave proposed anoer raio-pe esiaor for Y using sraified ranked se sapling as follows: W X =. ( ) =1 R RSS )3 RSS ] W ( RSS ) ( ) =1 Based on Upadaa and Sing [1] Mandowara and Mea [13] ave proposed wo ore raio-pe esiaors using bo coefficien of variaion and coefficien of kurosis of e auiliar variable in S RSS are given b () (3)

4 478 and Efficien Classes of Raio-Tpe Esiaors of Populaion Mean ( ) =1 R RSS )4 RSS ] W ( RSS ) ( ) C =1 W X C = W X C =. ( ) =1 R RSS )5 RSS ] W ( RSS ) C ( ) =1 (4) (5) Te Biases of R RSS )1 R RSS ) R RSS )3 R RSS )4 and R RSS )5 upo e firs order of approiaion are respecivel given b W pc pcc =1 r Bias( R RSS ) p ) Y (6) W p W ( i: ) p W ( i: ) W [ i: ] =1 r i=1 i=1 were p 1 5. and = = =1 WX =1W X C =1WX ( ) =1W X ( ) C and 5 3 = = =1 WX =1W X ( ) =1W X C ( ) =1W X C Te MSE s of R RSS )1 R RSS ) R RSS )3 R RSS )4 and R RSS )5 upo e firs order of approiaion are respecivel given b W C p C p C C =1 r R RSS ) p Y W W [ i: ] pw ( i: ) =1 r i=1 MSE( ) were p MEDIAN RANKED SET SAMPING ( MRSS ) In MRSS procedure selec rando saples eac of size fro e populaion and rank e unis wiin eac saple wi respec o a variable of ineres. If e saple size is odd fro eac saple selec for easureen e (( 1) / ) salles rank (i.e. e edian of e saple). If e saple size is even selec for easureen fro. (7)

5 Kan Sabbir and Kadilar 479 e firs / saples e ( / ) salles rank and fro e second / saples e (( ) / ) salles rank. Te ccle is repeaed r ies o ge r unis. Tese r unis for e MRSS daa. 4. STRATIFIED MEDIAN RANKED SET SAMPING Ibrai e al. [3] suggesed sraified edian ranked se sapling for esiaing e populaion ean. To esiae e finie populaion ean ( Y ) using SMRSS e procedure can be suarized as follows: Sep 1: Selec bivariae saple unis randol fro e srau of populaion. Sep : Arrange ese seleced unis randol ino ses eac of size. Sep 3: Te procedure of MRSS is en applied o obain independen MRSS saples eac of size o ge 1... = observaions. Here ranking is done wi respec o e auiliar variable X. Sep 4: Repea e above seps r ies o ge e desired saple size n = r. We use e following noaions for e SMRSS wen ranking is done wi respec o e auiliar variable X. For odd and even saple sizes e unis easured using S MRSS are denoed b S MRSSO and S MRSSE respecivel. For e j ccle and e srau e S MRSSO denoed b Y 1 X 1 Y 1 X 1... Y 1 X 1 1 j 1 j j j j j 1 j = 1... r and =1.... e [ ] = MRSSO i=1y 1 and 1 ( ) = MRSSO i=1x 1 i be e saple eans of Y and X in srau respecivel. For even saple size e Y X Y X... Y X 1 j 1 j j j j j i is SMRSSE is denoed b Y X Y 4 X 4... j j j Y X j j j

6 480 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean e 1 [ ] = MRSSE i=1 i i= i Y Y 1 and = X X i i= be e saple eans in e srau. ( MRSSE ) i=1 i To find Bias and MSE we define: [ ] ( ) 0( ) = S MRSSk Y S MRSSk X k and 1( k) = Y X suc a E( ik ( )) = 0 ( i = 01) were k = ( O E ) denoe e saple size odd and even respecivel. If saple size is odd we can wrie: 1 1 W W E 0( O) = =1 r Y E 1( O) = =1 r X 1 W E 0( O) 1( O) = =1. r XY If saple size is even we can wrie: W E 0( E) = =1 r Y W E 1( E) = =1 r X W E 0( E) 1( E) = =1. r XY

7 Kan Sabbir and Kadilar FIRST PROPOSED CASS OF ESTIMATORS We propose e following class of esiaors in W a X b = =1 R MRSSk ) p MRSSk ] W a( MRSS ) b =1 SMRSS given b (8) were a and b are known populaion paraeers wic can be coefficien of variaion coefficien of skewness coefficien of kurosis and coefficien of quariles of e auiliar variable. Also k = ( O E ) denoe e saple size odd and even respecivel. In ers of s we ave 1 R MRSSk ) Y 0 1 = (1 )(1 ) were = =1W X a =1W a X b R MRSSk ) ( Y ) Y.... (9) Taking epecaions we ge biases of ( ) respecivel given b Bias Bias R S MRSSk p 1 1 W MRSSO) p Y =1 r X YX MRSSE) p for odd and even saple sizes are (10) X W Y. (11) =1 r XY Squaring Equaion (9) and en aking epecaion e MSEs of ( ) odd and even saple sizes are respecivel given b R S MRSSk p for

8 48 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean MSE MSE (1) W MRSSO) p Y =1 r Y X XY R MRSSE) p Y Noe: i) If a =1 and b =0 en fro Equaion (8) we ge Y W =1 r X W Y. (13) =1 r XY X = MRSSk ) R MRSSk )1 MRSSk ] Te biases for odd and even saple sizes are respecivel given b (14) Bias Bias (15) 1 1 W MRSSO)1 Y =1 r X YX. (16) W MRSSE )1 Y =1 r X XY

9 Kan Sabbir and Kadilar 483 Te MSEs of R MRSSk )1 for odd and even saple sizes are respecivel given b MSE MSE (17) W =1 r Y X W Y. (18) =1 r XY W MRSSO)1 Y =1 r Y X XY R MRSSE )1 Y ii) If a =1 and b = C en fro Equaion (8) we ge W X C = =1 R MRSSk ) MRSSk ] W ( MRSS ) C =1 (19) were C is e populaion coefficien of variaion of e auiliar variable for e srau. Te biases of R MRSSk ) for odd and even saple sizes are respecivel given b Bias Bias (0) W Y. =1 r X XY (1) 1 1 W MRSSO) Y 1 1 =1 r X YX MRSSE) 1 1

10 484 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean Te MSEs of R MRSSk ) for odd and even saple sizes are respecivel given b MSE MSE () W MRSSO) Y 1 1 =1 r Y X XY W 1Y. (3) =1 r XY W R MRSSE) Y 1 =1 r Y X iii) If a =1 and b = ( ) en fro Equaion (8) we ge W X = ( ) =1 R MRSSk )3 MRSSk ] W ( MRSS ) ( ) =1 were ( ) is e populaion coefficien of kurosis of e auiliar variable for e srau. Te Biases of R MRSSk )3 for odd and even saple sizes are respecivel given b Bias Bias 1 1 W MRSSO)3 Y =1 r X YX (4) (5) W MRSSE)3 Y =1 r X XY. (6)

11 Kan Sabbir and Kadilar 485 Te MSEs of R MRSSk )3 for odd and even saple sizes are respecivel given b MSE MSE (7) W MRSSO)3 Y =1 r Y X XY W Y. (8) =1 r XY W R MRSSE)3 Y =1 r Y X iv) If a = ( ) and b = C en fro Equaion (8) we ge ( ) =1 R MRSSk )4 MRSSk ] W ( MRSS ) ( ) C =1 W X C = were ( ) and C are e populaion coefficien of kurosis and coefficien of variaion of e auiliar variable for e srau respecivel. Te Biases of R MRSSk )4 upo firs order of approiaion for odd and even saple sizes are respecivel given b Bias Bias 1 1 W MRSSO)4 Y 3 3 =1 r X YX (9) (30). (31) W MRSSE)4 Y 3 3 =1 r X XY

12 486 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean Te MSEs of R MRSSk )4 upo firs order of approiaion for odd and even saple sizes are respecivel given b MSE MSE (3) W MRSSO)4 Y 3 3 =1 r Y X XY W 3Y. (33) =1 r XY W R MRSSE)4 Y 3 =1 r Y X v) If a = C and b = ( ) en fro Equaion (8) we ge W X C =. ( ) =1 R MRSSk )5 MRSSk ] W ( MRSS ) C ( ) =1 Te Biases of R MRSSk )5 upo firs order of approiaion for odd and even saple sizes are respecivel given b Bias Bias 1 1 W MRSSO)5 Y 4 4 =1 r X YX (34) (35). (36) W MRSSE)5 Y 4 4 =1 r X XY

13 Kan Sabbir and Kadilar 487 Te MSEs of R MRSSk )5 upo firs order of approiaion for odd and even saple sizes are respecivel given b MSE MSE (37) W MRSSO)5 Y 4 4 =1 r Y X XY W 4Y. (38) =1 r XY W R MRSSE)5 Y 4 =1 r Y X 6. SECOND PROPOSED CASS OF ESTIMATORS Following Al-Oari [1] we proposed an oer class of raio-pe esiaors in S MRSS given b 1 3 =1 =1 MRSSk ) G MRSSk ] W ( MRSS ) q1 W ( MRSS ) q 3 =1 =1 W X q W X q = (1 ) (39) were is scalar quani and q 1 and q 3 are e firs and ird quariles of auiliar variable in e or In ers of srau respecivel. s we ave 1 1 MRSSk ) G Y = ( 1 ) 1 MRSSk ) G Y = Y ( 1 ) 1 ( 1 ) 0 1 were 1 = =1WX =1W X q1 = =1WX =1W X q3.

14 488 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean Te Biases of ( ) S MRSSk G sizes are respecivel given b and Bias Bias upo firs order of approiaion for odd and even saple 1 W ( ) 1 =1 r X S MRSSO G Y Y 1 S MRSSE G Y 1 W (40) =1 r YX W ( ) 1 =1 r X Y Te MSEs of ( ) S MRSSk G 1 saple sizes are respecivel given b W. (41) r XY =1 upo firs order of approiaion for odd and even and 1 1 W MSE MRSSO) G Y k( 1 ) =1 r Y X 1 W Y ( 1 ) (4) =1 r XY

15 Kan Sabbir and Kadilar 489 MSE MRSSE) G Te opiu value of is given b = W op =1 r Y W Y =1 r ( 1 ) X W Y ( 1 ). (43) =1 r XY i) For =1 in Equaion (39) we ge W X q =. 1 =1 MRSSk )6 MRSSk ] W ( MRSS ) q1 =1 (44) Te Biases of MRSSk )6 upo firs order of approiaion for odd and even saple sizes are respecivel given b and Bias Bias (45) 1 1 W MRSSO)6 1 1 =1 r X YX. (46) W MRSSE)6 Y 1 1 =1 r X XY

16 490 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean Te MSEs of MRSSk )6 upo firs order of approiaion for odd and even saple sizes are respecivel given b MSE MSE W MRSSO)6 Y 1 1 =1 r Y X YX (47) W 1Y. (48) =1 r XY W MRSSE)6 Y 1 =1 r Y X ii) For =0 in Equaion (39) we ge W X q =. 3 =1 MRSSk )7 MRSSk ] W ( MRSS ) q3 =1 Te Biases of MRSSk )7 upo firs order of approiaion for odd and even saple sizes are respecivel given b and Bias Bias 1 1 W MRSSO)7 =1 r X YX (49) (50). (51) W MRSSE)7 Y =1 r X XY

17 Kan Sabbir and Kadilar 491 Te MSEs of MRSSk )7 upo firs order of approiaion for odd and even saple sizes are respecivel given b MSE MSE (5) W MRSSO)7 Y =1 r Y X YX W 1Y. (53) =1 r XY W MRSSE)7 Y 1 =1 r Y X Table 1 PREs of Firs Class of Esiaors Wen Saple Size is Odd =3 = (34) = (346) = (111) = (111) W = ( ) = (555) and r =3. PRE (1) PRE () PRE (3) PRE (4) PRE (5)

18 49 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean Table PREs of Firs Class of Esiaors Wen Saple Size is Even =3 = (34) = (346) = (111) = (111) W = ( ) = (444) and r =3. PRE (1) PRE () PRE (3) PRE (4) PRE (5) Table 3 PREs of Second Class of Esiaors Wen Saple Size is Odd =3 = (34) = (346) = (111) = (111) W = ( ) = (555) and r =5. PRE (6) PRE (7) PRE( G )

19 Kan Sabbir and Kadilar 493 Table 4 PREs of Second Class of Esiaors Wen Saple Size is Even = 3 = (34) = (346) = (111) = (111) W = ( ) = (444) and r =5. PRE (6) PRE (7) PRE( G ) SIMUATION STUDY To copare e perforances of e proposed classes of esiaors a siulaion sud is conduced were ranking is perfored on e auiliar variable X. Bivariae rando observaions ( X( i) Y [ i] ) i= 1... ; and =1... are generaed fro a bivariae noral populaion aving paraeers. Using 0000 siulaions esiaes of MSEs for raio-pe esiaors are copued under S RSS and SMRSS. Esiaors are copared in ers of Percen Relaive Efficiencies ( PREs ). We used e following epressions o obain e PREs : MSE ( R RSS ) p PRE( p) = 100 k = ( O E) p = (1...5) MSE R MRSSk ) p PRE( s) = MSE R RSS )1 100 k = ( O E) s = (67 G) MSE MRSSk ) s

20 494 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean Te PREs of proposed classes of esiaors using SMRSS in coparison wi differen sraified ranked se esiaors for odd and even saple sizes are sown in Tables 1 3 and 4 respecivel. Te siulaion resuls sowed a wi decrease of e correlaion coefficiens PREs decreases wic are epeced resuls. Te nuerical values given in e firs eig rows are obained b assuing equal correlaions across e sraa wereas e las si rows assue unequal correlaions across e sraa. I is uc eas o conclude fro e resuls given in Tables 1 o 4 a our proposed esiaors perfor uc beer an eir copeiors. 8. NUMERICA IUSTRATION To observe perforances of e esiaors we used e following real daa se. Populaion [Source: Sing [18]] Te sud variable and e auiliar variable are defined below. : Te Tobacco producion in eric ons : Te area for Tobacco in specified counries during Table 5 Suar Saisics Srau 1 Srau Srau 3 N 1 = 1 N = 30 N 3 = 17 1 =3 =5 3 =3 n 1 =9 n = 15 n 3 =9 W 1 = W = W 3 = X 1 = X = X 3 = Y 1 = Y = Y 3 = R 1 = 1.97 R = 1.44 R 3 = 3.31 S = S = S = C = S = S = S = C =.3601 S = S = S = C = ( 1) = ( ) = ( 3) = = = =

21 Kan Sabbir and Kadilar 495 Fro above populaion we draw edian ranked se saples of odd and even saple sizes fro srau 1 s nd and 3 rd respecivel. Furer eac edian ranked se saple fro eac srau is repeaed wi nuber of ccles r. Hence saple sizes of sraified edian ranked se saples equivalen o sraified ranked se saple of sizes n = r. Te esiaed PREs based upon MSEs values of various sraified edian ranked se esiaors in coparison wi differen sraified ranked se esiaors are sown in Table 6. I sowed a our proposed raio-pe esiaors under S MRSS are ore efficien an eir copeiors in S RSS. Saple size Table 6 PREs of Differen Esiaors using Real Daa Se PRE (1) PRE () PRE (3) PRE (4) PRE (5) PRE (6) PRE (7) PRE( G ) Odd Even CONCUSION In is sud we proposed wo differen classes of raio-pe esiaors in SMRSS o esiae e finie populaion ean b adoping e Mandowara and Mea [13] and Al-Oari [1] esiaors. Te Biases and MSEs of ese proposed esiaors are derived up o firs order of approiaion. Bo siulaion and epirical sudies are conduced o observe e perforances of esiaors. On e basis of siulaion sud and nuerical illusraion our proposed raio-pe esiaors under S MRSS perfored beer as copared o respecive copeiive esiaors in esiaors MRSSk ) G is ore efficien. ACKNOWEDGMENT S RSS. Also aong all Te auors are ankful o e cief edior and e anonous referees for eir valuable coens wic elped o iprove e researc paper. REFERENCES 1. Al-Oari A.I. (01). Raio esiaion of e populaion ean using auiliar inforaion in siple rando sapling and edian ranked se sapling. Saisics and Probabili eers Al-Sale M.F. and Al-Kadiri M.A. (000). Double-ranked se sapling. Saisics and Probabili eers 48() Ibrai K. Sa M. and Al-Oari A.I. (010). Esiaing e Populaion Mean Using Sraified Median Ranked Se Sapling. Applied Maeaical Sciences 4(47)

22 496 Efficien Classes of Raio-Tpe Esiaors of Populaion Mean 4. Jeain A.A. and Al-Oari A.I. and Ibrai K. (007). Mulisage eree ranked se sapling for esiaion e populaion ean. Journal of Saisical Teor and Applicaion 6(4) Kadilar C. and Cingi H. (006). New raio esiaors using correlaion coefficien. InerSa Kadilar C. and Cingi H. (003). Raio esiaors in sraified rando sapling. Bioerical Journal 45() Kan. and Sabbir J. (016). A class of Harle-Ross pe unbiased esiaors for populaion ean using ranked se sapling. Haceepe Journal of Maeaics and Saisics 45(3) Kan. and Sabbir J. (016). Harle-Ross pe unbiased esiaors using ranked se sapling and sraified ranked se sapling. Nor Carolina Journal of Maeaics and Saisics Kan. and Sabbir J. (016). An efficien class of esiaors for e finie populaion ean in ranked se sapling. Open Journal of Saisics 6(3) Kan. and Sabbir J. (016). Iproved raio-pe esiaors of populaion ean in ranked se sapling using wo concoian variables. Pakisan Journal of Saisics and Operaional Researc 1(3) Kan. Sabbir J. and Gupa S. (016). Unbiased raio esiaors of e ean in sraified ranked se sapling. Haceepe Journal of Maeaics and Saisics Doi: /HJMS Kouncu N. (014). New difference-cu-raio and eponenial pe esiaors in edian ranked se sapling. Haceepe Journal of Maeaics and Saisics Doi: /HJMS Mandowara V.. and Mea N. (014). Modified raio esiaors using sraified ranked se sapling. Haceepe Journal of Maeaics and Saisics 43(3) Mclnre G. (195). A eod for unbiased selecive sapling using ranked ses. Crop and Pasure Science 3(4) Mulak H.A. (1997). Median ranked se sapling. journal of Applied Saisical Science Saawi H.M. and Mulak H.A. (1996). Esiaion of raio using ranked se sapling. Bioerical Journal 38(6) Saawi H.M. and Sia M.I. (003). Raio esiaion using sraified ranked se saple. Meron 61(1) Sing S. (003). Advanced Sapling Teor wi Applicaion How Micael Seleced A Vol. Neerlands. 19. Sisodia B.V.S. and Dwivedi V.K. (1981). A odified raio esiaor using coefficien of variaion of auiliar variable. Journal of Indian Socie of Agriculural Saisics Takaasi K. and Wakioo K. (1968). On unbiased esiaes of e populaion ean based on e saple sraified b eans of ordering. Annals of e Insiue of Saisical Maeaics 0(1) Upadaa.N. and Sing H.P. (1999). Use of ransfored auiliar variable in esiaing e finie populaion ean. Bioerical Journal 41(5)

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