Developing Efficient Ratio and Product Type Exponential Estimators of Population Mean under Two Phase Sampling for Stratification

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1 America Joural of Operatioal Researc 05 5: -8 DOI: 0.593/j.ajor Developig Efficiet Ratio ad Product Type Epoetial Eimators of Populatio Mea uder Two Pase Samplig for Stratificatio Subas Kumar adav S. S. Misra * Alok Kumar Sukla Departmet of Matematics ad Statiics A Cetre of Ecellece Dr. RM Avad Uiversity Faizabad Idia Departmet of Statiics D.A-V College Kapur Idia Abract Tis mauscript deals wit te eimatio of populatio mea of te mai variable uder udy usig auiliary iformatio uder two pase samplig for ratificatio. We te mai variable uder cosideratio is omogeeous ad te mea of te auiliary variable is ot kow double samplig is used to eimate te populatio mea of te mai variable. Stratified radom samplig is used we te mai variable uder udy is ot omogeeous. I te preset mauscript we ave proposed te efficiet ratio ad product type epoetial eimators uder two pase double samplig sceme for ratificatio for eimatig te populatio mea. Te epressios for te biases ad mea square errors of proposed eimators ave bee obtaied up to te fir order of approimatio. Te miimum mea square errors ave also bee obtaied for te proposed eimators. A compariso as bee made wit te eiig eimators of populatio mea i double samplig for ratificatio. A empirical udy is carried out to meet out te teoretical fidigs. Keywords Auiliary variable Double samplig Stratificatio Bias Mea square error. Itroductio Te proper use of auiliary iformatio icreases te efficiecies of te eimators of te populatio parameters. Tis iformatio is supplied by te auiliary variable X ad it is completely kow to te eperimeter. Te auiliary variable is igly positively or egatively correlated wit te mai variable uder udy. We X is igly positively correlated wit tat of ratio type eimators are used to eimate te populatio parameters wile product type eimators are used we X is igly egatively correlated wit oterwise regressio eimators are used to eimate te populatio parameters. Cocra [] utilized te positively correlated auiliary iformatio ad proposed te traditioal ratio eimator of populatio mea of udy variable. Robso [] proposed te traditioal product eimator of populatio mea usig egatively correlated auiliary iformatio. ater may autors icludig Upadyaya ad Sig [3] Sig [4] Sig ad Tailor [5] Sig et.al [6] [7] [8] Sig et al. [9] Kadilar ad Cigi [0] [] Tailor ad Sarma [] a ad Tia [3] adav [4] Padey et al. [5] Subramai ad Kumarapadiya [6] Solaki et al. [7] Oyeka [8] * Correspodig autor: sat_003@yaoo.co.i S. S. Misra Publised olie at ttp://joural.sapub.org/ajor Copyrigt 05 Scietific & Academic Publisig. All Rigts Reserved Jeelai et.al [9] adav ad Kadilar [0] etc. used auiliary iformatio for improved eimatio of populatio mea of te udy variable.. Metods ad Material et te fiite populatio cosi of diict ad idetifiable eterogeeous uits for te caracteriic uder udy. I tis situatio simple radom samplig is ot a appropriate tecique for eimatig populatio mea as it will ave a very large variace. To overcome tis problem we divide te wole populatio ito relatively omogeeous groups kow as rata. et te wole populatio is divided ito rata of size... wit W = as rata weigts. We tese rata weigts are ot kow two pase samplig sceme for ratificatio is used. Followig procedure as bee followed for two pase samplig sceme for ratificatio as give by Tailor et.al [] A sample of size is draw at fir pase usig simple radom samplig witout replacemet tecique ad te observatios are take o auiliary variable X oly. Tis sample of size is ratified ito rata based o auiliary variable. et be te umber of uits i t ratum... suc tat

2 Subas Kumar adav et al.: Developig Efficiet Ratio ad Product Type Epoetial Eimators of Populatio Mea uder Two Pase Samplig for Stratificatio =. 3 From tese uits a sample of size = is draw were 0< < is te predetermied probability of selectig a sample of size from rata of size ad it coitutes a sample of size =. From tis sample observatios o bot te variables ad X are take. Followig otatios ave bee used i tis mauscript = - Size of te sample at secod pase w = - t ratum weigt for te secod pase sample = yi - t ratum mea for te udy i = variable X = i - t ratum mea for te auiliary i = variable X Sy = yi - Populatio mea i= square deviatio for te udy variable S = i X - Populatio mea i= square deviatio for te auiliary variable X Sy = yi - t ratum mea i= square deviatio for te udy variable S = i X - t ratum mea i= square deviatio for te auiliary variable X Sy = yi i X - Covariace i= betwee y ad i t ratum ρ = S - Correlatio coefficiet betwee y y y SyS ad i te t ratum yds wy = = - Ubiased eimator of populatio mea i secod pase samplig ds w = = - Ubiased eimator of populatio mea X i secod pase samplig y yi i = samplig for te udy variable i i = samplig for te auiliary variable X i i= samplig for te auiliary variable X = - t ratum mea at secod pase = - t ratum mea at secod pase = - t ratum mea at fir pase f = - Fir pase samplig fractio Hase et.al [] proposed te classical combied ratio eimator for populatio mea uder ratified radom samplig as RC X = y. Ige ad Tripati [3] proposed te double samplig versio eimator of. as CRD = yds. ds Te combied product eimator of populatio mea i ratified radom samplig is defied as PC z Z = y.3 Te double samplig versio eimator of.3 is give by CPD = y ds zds z.4 were z is a auiliary variable wic is igly egatively correlated wit mai variable y uder udy ad z ds z ave teir usual meaigs. Te biases ad mea square errors of te eimators i. ad.4 respectively are B RS S W CRD = y y.5 X B CPD = Z ν W Syz.6

3 America Joural of Operatioal Researc 05 5: -8 3 were R MSE S W S R S R S f CRD = y + y + y.7 MSE S W S R S R S = Rz = ad W X Z f CPD = y + y + z z + z yz.8 =. Sig et.al [4] suggeed followig epoetial ratio ad product type eimators i ratified radom samplig based o Bal ad Tuteja [5] eimators of populatio mea uder simple radom samplig as W X X Re = y ep y ep = X + W X + W z Z z Z Pe = y ep y ep = z + Z W z + Z Tailor et.al [] proposed te double samplig eimators of te eimators i.9 ad.0 respectively as ds = yds ep + ds zds z = yds ep zds + z Te biases ad mea square errors of above eimators up to te fir order of approimatio are respectively were B 3S W 3S S f = y + y 8X.3 B W 4S RS = yz z z 8Z.4 β MSE S W S S f R y = y + y +.5 ν 4 R β MSE S W S S f Rz yz = y + y + z +.6 ν 4 R z z = i i= S z Z 3. Proposed Eimators i Syz = yi zi Z β = y S y = ρy. Sz Motivated by Tailor et.al [] ad Prasad [6] we proposed te followig ratio ad product type eimators as

4 4 Subas Kumar adav et al.: Developig Efficiet Ratio ad Product Type Epoetial Eimators of Populatio Mea uder Two Pase Samplig for Stratificatio ds = + ds κyds ep zds z = κ yds ep zds + z were κ ad κ are suitably cose coat to be determied suc tat te mea square errors of te eimators 3. ad 3. are miimum respectively. To udy te large sample properties of te proposed eimators we defie y e ds = + 0 ds X e = + ad = X + e suc tat Ee 0 = Ee = Ee = 0ad Ee S W S V yds f 0 = = y + y Ee S W S V ds f = = + X X f Ee = S X Cov yds ds f Eee 0 = = Sy + W Sy X X ν f Eee 0 = S y X f E ee = S X. Epressig i terms of ' O simplificatio we ave ei s we ave After epasio ad simplificatio we get X + e X + e = ep X + e + X + e κ e e e + e = ep + κ 0 0 = κ + e0 + + e e e e ee ee ee Subtractig from bot te sides ad takig epectatio bot sides we get bias of as 3 B Red = κ + Ee Ee Eee 0 + Eee 0 Eee Subtractig from bot te of 3.3 squarig ad simplifyig up to te fir order of approimatio we ave e e ee 0 ee 0 ee 3 = κ + e0 + + e e Epadig simplifyig ad takig epectatios o bot te sides we get te mea square error of order of approimatio as up to te fir

5 America Joural of Operatioal Researc 05 5: -8 5 { } MSE = κ + κ Ee 0 + Ee Eee 0 κ 3 Ee Ee Eee wic is miimum for + Ee0 + Ee Eee 0 E ee + Eee 0 3 Eee 0 Eee 0 E ee + Ee Ee A κ = = B were 3 Eee 0 Eee 0 E ee A = + Ee Ee f f + S + W S S 8 X 8 X = = f f f Sy W Sy Sy S + + X X 4X = B Ee Ee Eee f f + Sy + W Sy + S + W S ν X ν = f Sy W S + y X ν Ad te miimum mea square error is MSE mi A = B Similarly we ca get te bias ad mea square error of te eimator i 3. up to te fir order of approimatio as 3 B = κ + Ee Ee + Eee 0 Eee 0 Eee were zds = Z + e ad z = Z + e suc tat Ee = Ee = 0 ad Ee S W S V zds f = = z + z Z Z f Ee = S z Z Cov yds z ds f Eee 0 = = Sy + W Syz Z Z ν f Eee 0 = S yz Z f Eee = S z Z. 3.7

6 6 Subas Kumar adav et al.: Developig Efficiet Ratio ad Product Type Epoetial Eimators of Populatio Mea uder Two Pase Samplig for Stratificatio { } MSE = κ + κ Ee 0 + Ee + Eee 0 Eee 0 Eee 3 κ Ee Ee + Eee 0 Eee 0 Eee wic is miimum for 3 Eee 0 Eee 0 Eee Ee + Ee A κ = = B + Ee0 + Ee + Eee 0 Eee 0 Eee were 3 Eee 0 Eee 0 Eee A = Ee + Ee f 3 f Sz + W Sz + S z 8 Z 8 Z = = f f f Sy W Syz Syz S + + z Z Z 4Z = B Ee Ee Eee Eee Eee f f + Sy + W Sy Syz Z = f Sy W S + + yz Z ν Ad te miimum mea square error is MSE mi = B A Efficiecy Coditios From.7 ad 3.7 we ave MSE MSE > 0 If CRD mi S W S RS RS f A y + y + y > 0 4. B From.8 ad 3.7 we ave tat te proposed eimator is better ta te eimator MSE MSE > 0 i.e CPD mi From.5 ad 3.7 we ave S W S RS RS CPD if f A y + y + + y > 0 4. B

7 America Joural of Operatioal Researc 05 5: -8 7 MSE MSE > 0 If Te eimator mi β S W S S f R y A y + y + > ν 4 R B is better ta te eimator MSE MSE > 0 i.e mi if β S W S S f R y A y + y + + > ν 4 R B ote: Similar coditios are for te eimator. 5. Empirical Study We eamie te efficiecy of te proposed eimator over oter eimators. We cosider publised data sets wose atiics are give i Table. Te MSE ad percet relative efficiecy PRE values are give i Table. It is obviously see tat te proposed eimator are quite efficiet ta te oter eimators for te give Populatio. Table. Populatio Source: Gujarati [7] i Tousads of wildcats X i per barrel price Z i time Stratum I Stratum II Stratum I Stratum II S S z 5 5 S y S yz X S y Z S S y S z Table. MSE ad PRE Values of Eimators Eimators MSE PRE Eimators MSE PRE CRD CPD Pe d

8 8 Subas Kumar adav et al.: Developig Efficiet Ratio ad Product Type Epoetial Eimators of Populatio Mea uder Two Pase Samplig for Stratificatio 6. Results ad Coclusios I tis mauscript we proposed te improved ratio ad product type epoetial eimators of populatio mea uder two pase samplig usig te powerful kappa tecique. As we observe from te table- tat te proposed eimators ad ave lesser mea square errors ta te oter metioed eimators of populatio mea uder two pase samplig for ratificatio. It meas te eimated values of te parameter populatio mea by tese eimators are very close to te actual value of te parameter. Te mai aim of te udy is to fid te eimators wic predict te values very close to te actual value of te parameter wit miimum mea squared error. Here i te preset udy te major goal is obtaied as te proposed eimators ave te miimum mea square errors. Tus we ifer tat te proposed eimators sould be preferred for te eimatio of populatio mea i two pase samplig for ratificatio. REFERECES [] Cocra W.G. Te eimatio of te yields of te cereal eperimets by samplig for te Ratio of grai to total produce Jour. Agri. Sci [] Robso D.S. Applicatiof multivariate polykays to te teory of ubiased ratio type eimatio. J.Amer.at. Assoc [3] Upadyaya.. ad Sig H.P. Use of trasformed auiliary variable i eimatig te fiite populatio mea Biometrical Joural [4] Sig G.. O te improvemet of product metod of eimatio i sample surveys. JISAS [5] Sig H.P. ad Tailor R. Use of kow correlatio co-efficiet i eimatig te fiite populatio meas Statiics i Trasitio [6] Sig H.P. Tailor R. Tailor R. ad Kakra M.S. 004: A improved eimator of populatio mea usig power trasformatio Joural of te Idia Society of Agricultural Statiics [7] Sig H.P. Tailor R. Sig S. ad Kim J. M. A modified eimator of populatio mea usig power trasformatio. Statiical Papers [8] Sig H.P. Tailor R. ad Tailor R. 00. O ratio ad product metods wit certai kow populatio parameters of auiliary variable i sample surveys. SORT [9] Sig H.P. Tailor R. Tailor R. ad Kakra M.S. A improved eimator of populatio mea usig power trasformatio Joural of te Idia Society of Agricultural Statiics [0] Kadilar C. ad Cigi H. Ratio eimators i simple radom samplig Applied Matematics ad Computatio [] Kadilar C. ad Cigi H. A improvemet i eimatig te populatio mea by usig te correlatio co-efficiet Hacettepe Joural of Matematics ad Statiics Volume [] Tailor R. ad Sarma B.K A modified ratio-cum-product eimator of fiite populatio mea usig kow coefficiet of variatio ad coefficiet of kurtosis. Statiics i Trasitio-ew series [3] a Z. ad Tia B. 00: Ratio metod to te mea eimatio usig co-efficiet of skewess of auiliary variable ICICA Part II CCIS 06 pp [4] adav S.K. Efficiet eimators for populatio variace usig auiliary iformatio. Global Joural of Matematical Scieces: Teory ad Practical 3: [5] Padey H adav S K Sukla A.K. A improved geeral class of eimators eimatig populatio mea usig auiliary iformatio. Iteratioal Joural of Statiics ad Syems 6:-7. [6] Subramai J. ad Kumarapadiya G. 0: Eimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auiliary Variable. Iteratioal Joural of Probability ad Statiics 4: -8. [7] Solaki RS Sig HP Ratour A. A alterative eimator for eimatig te fiite populatio mea usig auiliary iformatio i sample surveys. Iteratioal Scolarly Researc etwork: Probability ad Statiics 0:-4. [8] Oyeka AC 0 Eimatio of populatio mea i poratified samplig usig kow value of some populatio parameters. Statiics i Trasitio-ew Series 3: [9] Jeelai M.I. Maqbool S. ad Mir S.A. Modified Ratio Eimators of Populatio Mea Usig iear Combiatio of Co-efficiet of Skewess ad Quartile Deviatio. Iteratioal Joural of Moder Matematical Scieces [0] adav S.K. ad Kadilar C. 03: Improved class of ratio ad product eimators Applied Matematics ad Computatio [] Tailor R. Coua S. ad Kim J-M. Ratio ad Product Type Epoetial Eimators of Populatio Mea i Double Samplig for Stratificatio Commuicatios for Statiical Applicatios ad Metods -9. [] Hase M. H. Hurwitz W.. ad Gurey M. Problems ad metods of te sample survey of busiess Joural of te America Statiical Associatio [3] Ige A. F. ad Tripati T. P. O doublig for ratificatio ad use of auiliary iformatio J. Id. soc. Agricul. ati. 39 p [4] R. Sig P. Caua. Sawa O liear combiatio of Ratio-product type epoetial eimator for eimatig fiite populatio mea Statiics i Trasitio [5] S. Bal ad R.K Tuteja Ratio ad product type epoetial eimator Iformatio ad optimizatio Scieces XII I [6] Prasad B. 989: Some improved ratio type eimators of populatio mea ad ratio i fiite populatio sample surveys Commuicatios i Statiics: Teory ad Metods [7] Gujrati D.. Basic Ecoometrics Fourt Editio McGraw-Hill.