A RATIO-CUM-PRODUCT ESTIMATOR OF POPULATION MEAN IN STRATIFIED RANDOM SAMPLING USING TWO AUXILIARY VARIABLES

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1 STATISTICA, ao XXII,. 3, 0 A RATIO-CUM-PRODUCT ESTIMATOR OF POPUATIO MEA I STRATIFIED RADOM SAMPIG USIG TWO AUXIIAR VARIABES R. Tailor, S. Coua, R. Tailor,. Garg. ITRODUCTIO A coutry or state frequetly requires estimates of agricultural productio to assess status of grai ad to make future policies regardig export ad import of grais accordig to eed. It requires te estimates of total productio, average productio ad per ectare productio of ay crop wic correspods to te problem of estimatio of populatio total, populatio mea ad ratio of two populatio meas respectively. Tis paper discusses te problem of estimatio of fiite populatio mea usig iformatio o two auxiliary variates. Auxiliary iformatio is ofte used by researcers i order to improve te efficiecies of estimators. Cocra (940) used auxiliary iformatio at estimatio stage ad evisaged ratio metod of estimatio tat provides ratio estimator. Ratio estimator as iger efficiecy we study variate ad auxiliary variates are positively correlated. Robso (957) developed product metod of estimatio tat provides product estimator. We study variate ad auxiliary variate are egatively correlated, product estimator gives iger efficiecy i compariso to simple mea estimator provided correlatio coefficiet betwee study variate ad auxiliary variate is greater ta alf of te ratio of coefficiet of variatio of auxiliary variate ad coefficiet of variatio of study variate. I bot, ratio ad product metods of estimatio, populatio mea of te auxiliary variate is assumed to be kow. Sig (967) utilized iformatio o two auxiliary variates, oe is positively correlated ad aoter is egatively correlated wit te study variate ad suggested ratio-cum-product estimator of populatio mea i simple radom samplig. ater may autors proposed various ratio ad product type estimators i simple radom samplig, for istace see Sisodia ad Dwivedi (98), Padey ad Dubey (988), Upadyaya ad Sig (999), Sig ad Tailor (003), Sig et al. (004), Sig ad Tailor (005), Kadilar ad Cigi (006), Sig et al. (009), Sig et al. (0), etc.. Hase et al. (946) defied combied ratio estimator usig auxiliary iformatio i stratified radom samplig. May autors icludig Kadilar ad Cigi (003, 005, 006) ad Sig et al. (008) worked out ratio type estimators i stratified radom samplig.

2 88 R. Tailor, S. Coua, R. Tailor,. Garg Simple radom samplig tecique as some sortcomigs like less represetative of differet sectios of te populatio, admiistrative icoveiece ad less efficiecy i case of eterogeeous populatio. iterature reveals tat ratiocum-product estimator performs better ta ratio ad product type estimators i simple radom samplig uder certai coditios. Tis motivates autors to work out Sig (967) ratio-cum-product estimator i stratified radom samplig ad study its properties. Cosider a fiite populatio U { U, U,..., U } of size ad it is divided ito strata of size (,,..., ). et be te study variate ad X ad Z be two auxiliary variates takig values yi, x i ad z i (,,..., ; i,,..., ) o i t uit of te t stratum. A sample of size is draw from eac stratum wic costitutes a sample of size defie: ad we X Z yi : stratum mea for te study variate, i xi : i zi : i t t stratum mea for te auxiliary variate X, t stratum mea for te auxiliary variate Z, y W : populatio mea of te study vari- i i X x W X i i ate, : populatio mea of te auxiliary variate X, Z z W Z y x : populatio mea of te auxiliary variate Z, i i yi : sample mea of te study variate for i xi : sample mea of te auxiliary variate X for i t stratum, t stratum,

3 A ratio-cum-product estimator of populatio mea etc. 89 z zi i : sample mea of te auxiliary variate Z for t stratum, W : stratum weigt of t stratum. Usual ubiased estimators of populatio meas, X ad Z i stratified radom samplig are defied respectively as y x z W y, () st W x, () st W z. (3) st I te lie of Cocra (940) ratio estimator, Hase et al. (946) utilized kow value of populatio mea X of auxiliary variate X ad defied combied ratio estimator for populatio mea as RC y X. (4) st x st Here it is assumed tat te study variate ad te auxiliary variate X are positively correlated. We te study variate ad te auxiliary variate Z are egatively correlated, assumig tat te populatio mea Z of auxiliary variate Z is kow, combied product estimator is defied as PC z Z. (5) st yst Te bias ad mea squared error of RC ad PC, up to te first degree of approximatio, are obtaied as B ( ) W ( RS S ), (6) RC x X yx

4 90 R. Tailor, S. Coua, R. Tailor,. Garg B ( ) W S, (7) PC Z yz RC y x yx MSE( ) W ( S R S R S ), (8) PC y z yz MSE( ) W ( S R S R S ) (9) were S y, y ( i ) i S x X, x ( i ) i S z Z, z ( i ) i S ( y )( x X ), yx i i i S ( y )( z Z ), yz i i i S ( x X )( z Z ), xz i i i R, R ad X Z.. PROPOSED ESTIMATOR Assumig tat te populatio meas of auxiliary variates X ad Z are kow, Sig (967) proposed a ratio-cum-product estimator for populatio mea as RP X z y x Z (0) were yi i xi i y ad x are ubiased estimates of populatio meas ad X i simple radom samplig witout replacemet. We propose Sig (967) ratio-cum- product estimator RP i stratified radom samplig as

5 A ratio-cum-product estimator of populatio mea etc. 9 WX Wz ST X z st RP yst W. () xst Z Wx WZ To compare te efficiecy of te proposed estimator i compariso to oter estimators, bias ad mea squared error of te proposed estimator are obtaied. To obtai te bias ad mea squared error expressios of te proposed estimator ST RP, we write y ( e ), x X ( e ) ad z Z ( e ) 0 suc tat Ee ( 0) Ee ( ) Ee ( ) 0 ad 0 C y E( e ), Cx E( e ), Cz E( e ), Ee ( e ) C C, ( ) 0 yx y x E e0e yzc ycz E( e e ) C C. xz x z Expressig () i terms of e ' s, we ave i WX WZ( e ) ST RP ( 0 ) WX ( e) WZ W e ST ( e )( e )( e ), () RP 0 were e 0 We 0, e WXe X ad e WZe Z suc tat Ee ( ) Ee ( ) Ee ( ) 0 ad 0 E( e ) W S, 0 y E( e ) W S, x X E( e ) W S, z Z

6 9 R. Tailor, S. Coua, R. Tailor,. Garg E( ee 0 ) W S yx, X E( ee 0 ) W S yz ad Z E( ee ) W Sxz. XZ After solvig () we get te bias ad mea squared error of proposed estimator upto te first degree of approximatio as ST S S yx Sxz S x yz B ( RP ) W, (3) X X XZ Z ST RP y x z yx yz xz MSE ( ) W { S RS RS ( RS RS RRS )}. (4) 3. EFFICIEC COMPARISOS To see te efficiecy of te proposed estimator i compariso to oter cosidered estimators, we compare te mea squared error of te proposed estimator wit variace or mea squared errors of oter estimators. Variace of usual ubiased estimator i stratified radom samplig y is st y V( y ) W S. (5) st Compariso of (4) ad (5) sows tat te proposed estimator ST RP would be more efficiet ta usual ubiased estimator y st if W { RSx RSz ( RS yx RS yz RRSxz )} 0. (6) From (8) ad (4), it is observed tat te proposed estimator ST RP would be more efficiet ta combied ratio estimator RC if W { RSz R( S yz RSxz )} 0. (7) Compariso of (9) ad (4) reveals tat te proposed estimator ST RP would be more efficiet ta combied product estimator PC if

7 A ratio-cum-product estimator of populatio mea etc. 93 W { RSx R( S yx RSxz)} 0. (8) Expressios (6), (7) ad (8) provide te coditios uder wic te proposed estimator ST RP would ave less mea squared error i compariso to mea squared error of usual ubiased estimator combied product estimator PC. y st, combied ratio estimator RC ad 4. EFFICIEC COMPARISOS I CASE OF PROPORTIOA AOCATIO t We te uits from te stratum are selected accordig to proportioal allocatio i.e. te,. I case of proportioal allocatio, variace of ubiased estimator y st, mea squared error of combied ratio estimator RC, combied product estimator PC ad ratio-cum-product estimator ST are obtaied as V( y ) st prop W S y RP, (9) MSE( ) W ( S R S R S ), (0) RC prop y x yx MSE( ) W ( S R S R S ), () PC prop y z yz MSE ( ) W{ S RS RS ST RP prop y x z ( RS RS RRS )}. yx yz xz () From (9), (0), () ad (), it is observed tat i case of proportioal allocatio, proposed estimator ST RP would be more efficiet ta (i) y st if W{ RSx RSz ( RS yx RS yz RRSxz)} 0, (3)

8 94 R. Tailor, S. Coua, R. Tailor,. Garg (ii) RC if W{ RSz R( S yz RSxz)} 0, (4) (iii) PC if W { RSx R( S yx RSxz )} 0. (5) Expressios (3), (4) ad (5) provides te coditios uder wic proposed estimator ST RP would ave less mea squared error i compariso to usual ubiased estimator y st, combied ratio estimator RC ad combied product estimator PC i case of proportioal allocatio. 5. EMPERICA STUD To see te performace of te proposed estimator empirically i compariso to oter estimators, we cosider two atural populatio data sets. Descriptio of te populatios are give below. Populatio I [Source: Murty (967)] : Output, X : Fixed capital, Z : umber of workers. =0 =5 = =3 =5 =5 = =35.60 X =4.40 X = Z =5.80 Z =60.60 S y =65.9 S x =74.87 S yx = S zx =38.08 S x =66.35 S yx = S zx =87.9 S z =0.75 S yz =4.6 S y = S z =4.84 S yz =536.4 Populatio II [Source: atioal Horticulture Board (00)] : Productivity (MT/Hectare), X : Productio i 000 Tos, Z : Area i 000 Hectare.

9 A ratio-cum-product estimator of populatio mea etc. 95 =0 =7 =3 =4 =0 =0 =.70 =3.67 X =0.4 X =309.4 Z =6.0 Z =80.67 S y =0.54 S x =3.53 S yx =.60 S zx =.75 S x =80,54 S yx =83.47 S zx =68.57 S z =.9 S yz =-0.0 S y =.4 S z =0.8 S yz =-7.06 Estimators TABE Empirical presetatio of coditios (6), (7) ad (8) uder wic te suggested estimator ST RP is more efficiet ta y, st RC ad PC y st RC Populatio < < < 0 Populatio < < < 0 PC TABE Percet relative efficiecies of y st, RC ad ST RP wit respect to y st, PC Estimators y st RC Populatio Populatio PC ST RP Estimators TABE 3 Percet relative efficiecies of y st, RC, PC ad ST RP wit respect to y st (i case of proportioal allocatio) y st Populatio Populatio RC PC ST RP 5. COCUSIO Sectio 3 provides te coditios uder wic te proposed estimator ST RP as less mea squared error i compariso to usual ubiased estimator y, combied ratio estimator RC ad combied product estimator PC. Sectio 4 tat deals wit te efficiecy comparisos i case of proportioal allocatio, provides te coditios (3), (4) ad (5) uder wic proposed estimator ST RP is more efficiet ta usual ubiased estimator, combied ratio estimator ad combied product estimator i case of proportioal allocatio. Te coditios give i sectio 3 ave bee cecked empirically ad give i st

10 96 R. Tailor, S. Coua, R. Tailor,. Garg Table, wic sows tat all coditios are satisfied for bot te populatios. Table exibits tat te proposed estimator ST RP as igest percet relative efficiecy i compariso to y st, RC ad PC. Terefore, it is cocluded tat proposed estimator ST RP is more efficiet ta y st, RC ad PC provided tat coditios (6), (7) ad (8) are satisfied Table 3 sows tat i case of proportioal allocatio, te proposed estimator as igest percet relative efficiecy i compariso to oter cosidered estimators i bot populatios. It is importat to ote tat proportioal allocatio provides iger percet relative efficiecy as compared to o-proportioal case. Tus te proposed estimator ST RP is recommeded for use i practice istead of oter covetioal estimators we coditios give i sectio 3 ad 4 are satisfied. Tis study uses iformatio o te populatio mea of two auxiliary variates. Te same study may be exteded usig various kow parameters of auxiliary variates suc as coefficiet of variatio, coefficiet of kurtosis, correlatio coefficiet betwee two auxiliary variates etc. see Sig ad Tailor (005). Usig simulatio study impact of use of various parameters of auxiliary variates ca also be studied i te estimatio of populatio parameters. Scool of studies i statistics, Vikram Uiversity Ujjai Departmet of Statistics, Sri Vaisav Istitute of Maagemet Wood Properties ad Uses Divisio, Istitute of Wood Sciece ad Tecology, Bagalore Scool of Scieces Idira Gadi atioal Ope Uiversity (IGOU) RJESH TAIOR SUI CHOUHA RITESH TAIOR EHA GARG ACKOWEDGEMET Autors are takful to te referee for is valuable suggestios regardig te improvemet of te paper. REFERECES B.. PADE, V. DUBE, (988), Modified product estimator usig coefficiet of variatio of auxiliary variable, Assam Statistical Review,, pp B.V.S. SISODIA, V.K.. DWIVEDI, (98), A modified ratio estimator usig coefficiet of variatio of auxiliary variable, Joural of Idia Society of agricultural Statistics, 33,, pp C. KADIAR, H. CIGI, (003), Ratio estimators i stratified radom samplig, Biometrical Joural, 45,, pp. 8-5.

11 A ratio-cum-product estimator of populatio mea etc. 97 C. KADIAR, H. CIGI, (005), A ew Ratio Estimator i Stratified Radom Samplig, Commuicatios i Statistics-Teory ad Metods, 34, pp C. KADIAR, H. CIGI, (006), A improvemet i estimatig te populatio mea by usig te correlatio coefficiet, Hacettepe Joural of Matematics ad Statistics, 35,, pp D.S. ROBSO, (957), Applicatio of multivariate polykays to te teory of ubiased ratio-type estimatio, J. Amer. Statist. Assoc., 5, pp H.P. SIGH,.. UPADHAA, R. TAIOR (009). Ratio-cum-product type expoetial estimator. Statistica, 69, 4, pp H.P. SIGH, R. TAIOR, (003), Use of kow correlatio coefficiet i estimatig te fiite populatio mea, Statistics i Trasitio, 6, 4, pp H.P. SIGH, R. TAIOR, (005), Estimatio of fiite populatio mea usig kow correlatio coefficiet betwee auxiliary caracters, Statistica, ao 65, 4, pp H.P. SIGH, R. TAIOR, R. TAIOR, M.S. KAKRA, (004), A improved estimator of populatio mea usig power trasformatio, Joural of Idia Society of Agricultural Statistics, 58,, pp H.P. SIGH, R. TAIOR, S. SIGH, ad J.M. KIM, (008), A modified estimator of populatio mea usig power trasformatio, Statistical Papers, 49, pp H.P. SIGH, R. TAIOR, ad R. TAIOR (0). Estimatio of fiite populatio mea i two - pase samplig wit kow coefficiet of variatio of a auxiliary caracter. Accepted for publicatio i Statistica... UPADHAA, H.P. SIGH, (999), Use of trasformed auxiliary variable i estimatig te fiite populatio mea. Biometrical Joural, 4, 5, pp M.. MURTH, (967), Samplig teory ad metods, Statistical Publisig Society, Calcutta, Idia, pp. 8. M. P. SIGH, (967), Ratio cum product metod of estimatio, Metrika,, pp M.H. HASE, W.. HURWITZ, ad M. GURE, (946). Problems ad metods of te sample survey of busiess, Joural of America Statistical Associatio. Amer. Statist. Assoc., 4, pp ATIOA HORTICUTURE BOARD (00). ttp://b.gov.i/statistics/area-productiostatistics.tml W.G. COCHRA, (940), Te estimatio of yield of cereal experimets by samplig for te ratio of gai to total produce, Joural of Agricultural Sciece., 30, pp SUMMAR A ratio-cum-product estimator of populatio mea i stratified radom samplig usig two auxiliary variables Tis paper proposes a ratio-cum-product estimator of fiite populatio mea i stratified radom samplig usig iformatio o populatio meas of two auxiliary variables. Te bias ad mea squared error expressios are derived uder large sample approximatios. Proposed estimator as bee compared wit usual ubiased estimator i stratified samplig, combied ratio estimator ad combied product estimator teoretically as well as empirically.