REVISTA INVESTIGACION OPERACIONAL VOL. 35, NO. 1, 4957, 2014


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1 EVISTA IVESTIGAIO OPEAIOAL VOL. 35, O., 957, 0 O A IMPOVED ATIO TYPE ESTIMATO OF FIITE POPULATIO MEA I SAMPLE SUVEYS A K P Swai Former Professor of Statistics, Utkal Uiversit, Bhubaeswar7500, Idia ABSTAT I this paper a alterative ratio tpe expoetial estimator is suggested ad is compared with Bahl ad Tuteja s ratio tpe expoetial estimator ad classical ratio estimator as regards bias ad mea square error with large sample approximatios both theoreticall ad with umerical illustratio. KEYWODS: Simple radom samplig, ratio tpe estimators, Bias, Mea square error. MS: 6D05 ABSTAT I this paper a alterative ratio tpe expoetial estimator is suggested ad is compared with Bahl ad Tuteja s ratio tpe expoetial estimator ad classical ratio estimator as regards bias ad mea square error with large sample approximatios both theoreticall ad with umerical illustratio.. ITODUTIO: The use of auxiliar iformatio dates back to ear 93, whe ema used it for stratificatio of the fiite populatio. ochra(90 used auxiliar iformatio i estimatio procedure ad proposed ratio method of estimatio to provide more efficiet estimator of the populatio mea or total compared to the simple mea per uit estimator uder certai coditios whe the auxiliar variable has positive correlatio with the stud variable i questio. Let U = (U, U U be the fiite populatio of size. To each uit U i (i=,, i the populatio paired values ( i, x i correspodig to stud variable ad a auxiliar variable x, correlated with are attached. ow, defie the populatio meas of the stud variable ad auxiliar variable x as Y i, X x i i i Thus, the populatio ratio is defied as Y X Further, defie the fiite populatio variaces of ad x ad their covariace as S, i Y S x, xi X ad S x i Y xi X, respectivel. i i Thus, the squared coefficiets of variatio of ad x ad their coefficiet of covariatio are respectivel defied b 0, 0 ad,where r s rs ( i Y ( xi X, rs, 0,,,... i Defie as the correlatio coefficiet betwee ad x i the populatio. A simple radom sample s of size is selected from U without replacemet ad the values ( i, x i, i=, are observed o the sampled uits. Assume that ad x are positivel correlated. The simple mea is a ubiased estimator of the populatio meay ad variace of is give b Var( 0 Y The classical ratio estimator of the populatio mea Y,usig auxiliar iformatio o x is give b i 9
2 X, x (. where ad x are sample meas of ad x respectivel, defied b The ratio estimator to O(/ are give b(see Sukhatme ad Sukhatme,970 i i ad x x i i evisages advace kowledge of X. The bias( B ad mea square error ( MSE of B( Y ( ( where r s, rs ( i Y ( xi X, r, s 0,, i The ratio estimator is a biased estimator ad the bias decreases with icrease i sample size. I large samples is more efficiet tha the simple mea per uit estimator if k 0 0 MSE( Y ( (.3 ( or if whe 0 0. Bahl ad Tuteja(99 have suggested a ratio tpe expoetial estimator give b X x BT exp( X x To O(/,the bias ad mea square error of BT 3 B( BT Y ( 0 8 (. are give b MSE( BT Y ( 0 0 (.6 BT is more efficiet tha ad 3 if k, (Bahl ad Tuteja,99: Upadhaa,Sigh,hatterjee ad Yadav (0 have show that to O(/,Bahl ad Tuteja s(99 ratio tpe expoetial estimator is more efficiet tha the classical ratio estimator whe ad x follow bivariate ormal distributio, coefficiets of variatio of ad x are equal ad 3 0. I the followig we suggest a ratio tpe estimator with square root trasformatio ad compare it with the Bahl Tuteja s (99 ratio tpe expoetial estimator ad classical ratio estimator both theoreticall ad with the help of some atural ad artificial populatios.. A ATIO TYPE ESTIMATO USIG SQUAE OOT TASFOMATIO Defie the ratio tpe estimator usig square root trasformatio as. (.5 50
3 X x.. Bias ad Mea square error of SQ / SQ ( (. Y ( e ad x X ( e with E( e0 E( e 0, V ( e0 0 Write 0 V ( e 0 ad ov( eo, e. 5, i power series uder the assumptio that e for all possible samples ad retaiig terms Expadig SQ up to fourth degree i e 0 ad e,we have SQ Y Y ( e0 e e e e e0e 3 e0e 5 e0e ( ( SQ Y Y ( e0 e e0e ee e0e e0e e0e e e ( Thus, B ( SQ Y E( e0 E( e E( e E( e E( e E( e0e 3 5 E( e0e E( e0e (. 8 8 MSE( SQ E( SQ Y Y E( e0 E( e E( e0e Y E( e0e e0e E( e0e E( e0e E( e E( e ( ow, ( ( E( e ( ( ( 3 ( ( ( ( ( E( e0e ( ( ( 3 ( ( ( ( ( E( e e ( 3 ( ( ( 3 ( ( ( 3 3 ( ( E( e 03 ( ( ( ( ( ( E( e0e E( e0e ( ( ( ( where r s r s rs ( i Y ( xi X / Y X i, ( rs,,,3,.
4 (See Sukhatme,Sukhatme ad Asok (98 Hece, 3 B ( SQ Y ( 0 8 A 3 5 B 05 5 Y ( 03 ( D 05 5 ( ( MSE( SQ Y ( 0 0 A( 03 8 ( Y B( 0 3 D( where A B 6 6 ( ( 3.. Bias ad Mea square error of BT D ( ( ( ( 3 Expadig BT i power series with the same assumptios used i expadig SQ ad retaiig terms up to ad icludig degree four i e 0 ade,we have e ee BT Y Y ( e0 e e e e0e e0e ( BT Y Y ( e0 e e0e e0e e0e e e e0e e0e B ( BT Y ( 0 8 A 3 3 B 73 3 Y ( 03 ( D 73 3 ( MSE( BT E( BT Y Y E( e e e e Y E( e0e e0e e e e0e e0e Y ( 0 0 A( B( 0 3 D( ( Bias ad Mea Square Error of 5 (.8
5 We fid (see Sukhatme, Sukhatme ad Asok,98 to secod order approximatio B( Y ( 0 Y A ( 03 B ( D ( 0 0 (.0 MSE( Y ( 0 0 A( (. Y B( 0 3 D( Uder the assumptio of bivariate ormalit of ( x, ad equalit of coefficiets variatio of ad x,i.e. it ma be show that MSE( Y ( ( 8 6 ( MSE( BT Y ( ( 6 8 ( MSE( SQ Y ( ( 6 8 (. 3. OMPAISO OF EFFIIEY (i osiderig terms up to O(/, ow, MSE( Y ( 0 0 MSE( SQ MSE( BT Y ( 0 0 B = B( Y ( 0 B = 3 B( SQ B( BT Y ( 0 8 MSE( MSE( MSE( MSE( 3 Y ( 0 SQ BT Thus, to O(/, BT ad more efficiet tha if ad SQ are equall efficiet. The are more efficiet tha 3 0 or 3 k, if k. Hece, SQ ad BT are more efficiet tha both Further, to O(/ the biases of SQ ad BT is either greater tha 5/ or less tha /.. (ii To O(/,for large, ad if. are equal ad further the are less biased tha if k 53
6 MSE( MSE( Hece, SQ ( ( ( 33 3 ( 9 9 to 3 BT E e E e0e SQ will be more efficiet tha BT (/ O 0 or 0 if 0 0 or whe 0 0 Uder bivariate ormalit of ad x with the assumptio that 0 0 betwee ad x equal to we have MSE( BT MSE( = Y ( 0 0 ad the correlatio coefficiet Y 0 ( ( 6 8 (see Upadhaa et al,0 ad correct the coefficiet of as 3/8 istead of 03/8 MSE( SQ MSE( = Y 0 ( ( If, both MSE( BT MSE( ad MSE( SQ MSE( O(/ the sufficiet coditio that both BT ad SQ 3 0 Further, SQ will be more efficiet tha BT if Hece, SQ will be more efficiet tha BT ad if 3 are egative.hece, to are more efficiet tha is that To sum up uder the assumptios of bivariate ormalit of ad x ad the equalit of coefficiets of variatio of ad x,the preferred estimators for rages of re show below i Table. Table ompariso of Estimators Preferred estimator 0 3 SQ BT 5
7 3. UMEIAL ILLUSTATIO: Illustratio osider atural populatios described i Table to compare the ratio tpe estimator with square root trasformatio estimator with classical ratio estimator as regards bias ad mea square error.bias ad mea square error have bee computed exceptig the commo costats(table3: Pop o. Table atural Populatios Descriptio x Sampford (96 7 Acreage uder oats i 957 Sigh ad haudhar(986 Acreage of crops ad grass i 97 6 Area uder wheat! Total cultivated area durig x x Koij(973 6 Food expediture Total expediture Murth(967 6 Area uder witer padd(i acres 5 Sigh ad haudhar(986 7 o. of milch aimals i surve Murth(967 6 Output for factories(000s 7 Pase ad Sukhatme(967 8 Pase ad Sukhatme(967 9 Sigh ad haudhar((986 0 Pase ad Sukhatme(967 Pase ad Sukhatme(967 Geographical area (i acres o. of milch aimals i cesus976 6 Proge mea(mm Paretal plat value(mm 0 Paretal plot mea(mm Paretal plat value(mm 5 Area uder wheat i 973 Area uder wheat i 97 5 Paretal plot mea(mm Paretal plat value(mm 0 Proge mea(mm Paretal plat value(mm Swai(003 9 o. of milch cows i Fixed capital(000s o. of milch cows cesus Table 3 ompariso of Bias ad Mea square Error Pop. o ( Bias SQ Y Bias( Y ( MSE SQ Y 55 MSE( Y MSE( 00 MSE( SQ
8 ommets : For the populatios uder cosideratio,the ratio tpe estimator with square root trasformatio is less biased tha the classical ratio estimator.further, For populatios 5, the ratio tpe estimator with square root trasformatio is more efficiet tha the classical ratio estimator i the sese of havig lesser mea square error. Illustratio osider a hpothetical populatio of size =5 as follows: : x : For a simple radom sample (without replacemet of size =,the exact biases ad mea square errors of SQ ad BT are give i Table., Table : ompariso of Biases ad Mea square errors Estimators Absolute Bias M. S.E elative Efficiec SQ BT ommets: As per Illustratio (ithe ratio tpe estimator with square root trasformatio ad ratio tpe expoetial estimator have earl equal bias ad efficiec ad further the are less biased ad more efficiet tha the classical ratio estimator.. (ii The classical ratio estimator is less efficiet tha the simple mea per uit estimator,where as the ratio tpe estimator with square root trasformatio ad the ratio tpe expoetial estimator are more efficiet tha. 5. OLUSIOS: (ito O(/ SQ ad BT are equall efficiet ad more efficiet tha both ad if 3 k. To same order of approximatio both SQ ad BT are less biased tha if k is either 5 greater tha or less tha. (ii Uder the assumptio of bivariate ormalit of ( x, ad equalit of coefficiets of variatio of ad x,to O(/ SQ is more efficiet tha both ad BT for 3. However, for the values of 3 i the rage, is to be preferred over, BT. SQ ad 56
9 (iii umerical illustratio with the help of twelve atural populatios shows that there might arise situatios whe SQ is less biased ad more efficiet tha uder large sample approximatios. Also, for the artificial populatio uder cosideratio differece betwee SQ happes to be less biased ad more efficiet tha. However, the SQ ad BT as regards bias ad mea square error is ol margial. EEIVED APIL, 03 EVISED OVEMBE, 0 EFEEES BAHL,S. ad TUTEJA,.K. (99: atio ad Product Tpe Expoetial Estimator.,Iformatio ad Optimizatio Scieces,, OHA,W.G.(90:The estimatio of the ields of cereal experimets b samplig for the ratio of grai to total produce. Jour.Ag.Sc.,30, KOIJ,H.S.(973: Statistical Theor of Sample Surve Desig ad Aalsis. orth Hollad Publishig o. Amsterdam. MUTHY,M..(967: Samplig: Theor ad Methods. Statistical Publ. Societ,ISI, alcutta. 5 EYMA, J. (93:O the two differet aspects of the represetative method: The method of stratified samplig ad the method of purposive selectio. J. oal Stat. Soc.,97, PASE,V.G. ad SUKHATME,P.V.(967: Statistical Methods for Agricultural Workers. Idia oucil of Agril.es, ew Delhi 7 SAMPFOD,M..(96:A Itroductio to Samplig Theor. Oliver ad Bod. Lodo. 8 SIGH,D ad HAUDHAY,F.S.(986:Theor ad Aalsis of Sample Surve Desigs, First Editio,Wile Easter Ltd. alcutta. 9 SUKHATME, P.V. ad SUKHATME, B.V.(970: Samplig Theor of Surves with Applicatios, Asia Publishig House, alcutta. 0 SUKHATME, P.V., SUKHATME, B.V. ad ASOK,.(98:Samplig Theor of Surves with Applicatios,Idia Societ of Agricultural Statistics, ew Delhi. SWAI, A.K.P..(003: Fiite Populatio Samplig: Theor ad Methods. South Asia Publishers, ew Delhi. UPADHYAYA,L..,SIGH,H.P.,HATTEJEE,S. ad YADAV,.(0: Improved atio ad Product Expoetial Tpe Estimator, Jour. Stat.Theo. ad Pract. 5,
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