Optimal Search for Efficient Estimator of Finite Population Mean Using Auxiliary Information

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1 America Joural of Operatioal Reearch 04, 4(): 8-34 DOI: 0.593/j.ajor Optimal Search for Efficiet Etimator of Fiite Populatio Mea Uig Auiliary Iformatio Subhah Kumar Yadav, S. S. Mihra,*, Suredra Kumar Departmet of Mathematic ad Statitic, Dr. RML Avadh Uiverity, Faizabad, U.P., Idia Departmet of Mathematic, Govt. Degree College, Pihai, U.P., Idia Abtract I thi maucript, we preet the optimal earch for efficiet etimator of fiite populatio mea uig auiliary iformatio. Thi eek to develop efficiet ratio ad product type epoetial etimator of populatio mea uder predictive method of etimatio utilizig auiliary iformatio. The large ample propertie of the propoed etimator have bee tudied up to the firt order of approimatio. The epreio for the biae ad mea quare error (MSE) have bee obtaied up to the firt order of approimatio. The miimum value of the MSE of propoed etimator have alo bee obtaied for the optimum value of the characterizig calar kow a kappa. A compario ha bee made with previou etimator till optimal etimator havig leat MSE ha bee obtaied leadig to highet efficiecy which ha bee demotrated o umerical data uder tudy. Keyword Efficiet Etimator, Predictive Etimatio, Fiite Populatio Mea, MSE, Efficiecy. Itroductio The ytem of upplemetary populatio iformatio i modeled to rejig the etire pectrum of etimatio proce i the theory ad practice of ample urvey. Preetly, it i widely ued by theorit ad practitioer of etimatio kowledge of populatio parameter of the give populatio. Availability of variou approache to cotruct improved efficiet etimator for populatio parameter i ot oly the eceary coditio but traiig of mid to them i alo ufficiet. Why do we ue the theory of predictio i a quiteetial apect of echatmet for the reearcher egaged i thi field of ivetigatio. The awer to thi quetio of attractio of the method lie i the fact that it provide u geeral framework for tatitical iferece o the character of fiite populatio, vide Cochra (977). I the theory of urvey amplig, it i uaimouly accepted that the uitable ue of auiliary iformatio improve the efficiecy of the etimator of the parameter of the populatio uder coideratio for the mai characteritic (y) uder tudy. The auiliary variable () i the variable about which we have full iformatio ad which i highly poitively or egatively correlated with the mai variable uder tudy. Whe the auiliary variable i highly poitively correlated with the mai variable, the the ratio method of etimatio i ued i which the ratio type * Correpodig author: at_003@yahoo.co.i (S. S. Mihra) Publihed olie at Copyright 04 Scietific & Academic Publihig. All Right Reerved etimator are ued for the etimatio of the parameter. O the other had product type etimator uder product method of etimatio are ued for the etimatio of the populatio parameter whe the auiliary variable i highly egatively correlated with the mai variable uder tudy; vide Murthy (967) ad Da (988). May author ued auiliary iformatio for improved etimatio of populatio mea through ratio ad product type etimator for the mai characteritic uder dicuio. The latet referece ca be made of Ya ad Tia (00), Yadav (0), Padey et al. (0), ayak ad Sahoo (0), Subramai ad Kumarapadiya (0), Solaki et al. (0), Oyeka (0), Jeelai et al. (03), Saii (03), Yadav ad Kadilar (03), Sigh et al. (04) etc.. Optimal earch for olutio by uig computatioal method have bee widely ued i boilig dow the umerical compleity icludig ivetory populatio i upply chai maagemet etc for attaiig efficiet etimator of performace meaure of ivetory ytem vide for eample Mihra ad Sigh (0, 0, 03), Mihra ad Mihra (03) ad Yadav et al.(0, 04). The freh attempt ha bee made to preet optimal earch for efficiet etimator of fiite populatio mea uig auiliary iformatio i which mot efficiet ratio ad product type epoetial etimator have bee developed. Optimal earch cotiue ule data computatio provide u better efficiecie a compared to previouly eitig etimator. I the propoed etimator, kappa techique ha bee ued which eek to itroduce a cotat ad further the optimal value i obtaied by miimizig the mea quare error of the etimator. et, after puttig thi optimal value,

2 America Joural of Operatioal Reearch 04, 4(): miimum value of mea quare error i obtaied. Latly, reult dicuio ad cocluio are give i the lat ectio.. Material ad Method U ( UU,,..., U ) Let the fiite populatio coit of ditict ad idetifiable uit. Let the mai variable uder tudy i deoted by Y ad the auiliary variable by X. Thu ( Y, X ), ( i,,..., ) deote the i th i i obervatio for the mai ad auiliary variable repectively. Thu we have Y yi, the populatio mea of the mai variable i to be etimated. Further let S deote the et of all poible ample of fied deote the effective ize from the populatio U. Let ϑ() ample ize that i the umber of ditict elemet i ad deote the collectio of all thoe uit of U which are ot i. We deote, Y Y ϑ() i [ ϑ() ] y y i i. I predictive etimatio a pecific model i coidered to predict the o ampled value of the populatio. The predictio theory i baed o the model-baed theory. It coider a geeral framework for tatitical iferece to be draw for the parameter of the populatio i quetio. The geeral predictio theory coider differet ratio, product ad regreio type etimator of populatio parameter a the predictive etimator i.e, predictor of the uoberved uit of the populatio uder ome pecific model. Several author ued ratio, product ad regreio type etimator a predictor of populatio parameter i predictive etimatio theory. I predictive etimatio theory it i well kow that the ue of ratio ad product type etimator a predictor of the parameter uder coideratio of the uoberved uit of the populatio reult i the correpodig etimator of the populatio parameter for the whole populatio. For a pecific ample S, the populatio mea Y ca be writte a, ϑ() [ ϑ() ] Y Y + Y (.) The ample mea of the ample of ize (i.e. ϑ () ) i imple radom amplig i, y y i i that i Y i y. Thu, the populatio mea Y i () may be rewritte a, ( ) Y Y + Y (.) The uitable etimator for Y uig (.) ca be coidered a, ( ) t y+ T (.3) where T i the predictor of the populatio mea Y of uoberved uit of the populatio. Srivatava (983) coidered the followig etimator a the predictor of Y a, T y i y i, y T yr X, where, i i, X i i ( X ) ad X ( ) i i. ( ) He ha how that wheever thee etimator are ued a predictive etimator of Y, the the etimator t give i (.3) reult i correpodig claical etimator, X y y i i ad y R y repectively. However, he ha how that for the product predictive etimator, T yp y of Y, the etimator doe X ot reult i the claical product etimator yp y X of populatio mea Y. Thu, for T yp y, we have X X + ( ) t t ( X ) p (.4) The biae ad the mea quare error of the etimator y R, y P ad t p, up to the firt order of approimatio repectively are, ia( yr) θyc( C),

3 30 Subhah Kumar Yadav et al.: Optimal Search for Efficiet Etimator of Fiite Populatio Mea Uig Auiliary Iformatio ia( y ) P θycc, ia( t ) θyc [ C + f ( f ) ], p R θ y MSE( y ) Y [ C + C ( C)], (.5) P p θ y MSE( y ) MSE( t ) Y [ C + C (+ C)] (.6) where, θ ( f ), f ( ), y ( y ), C S Y C ( S X ), C ρ ( C C ), ρ ( S S S ), y y y Sy ( ) ( yi Y), S ( ) ( i X) i i ad Sy ( ) ( yi Y)( i X). i Sigh et al. (04) uggeted two ratio ad product type epoetial etimator of populatio mea Y uig ahl ad Tuteja (99) ratio ad product type epoetial etimator of populatio mea a the predictive etimator of Y repectively a, X Re ep t t y+ y X + ( ) y yep X + (.7) ( X ) X t t ep Pe y+ y + X ( ) y yep X + (.8) X + ( ) The biae ad the mea quare error of above etimator up to the firt order of approimatio repectively are, θ ia( tre) YC [3 4( C + f )], 8 θ ia( tpe) YC 4C 8 ( f ), C MSE( tre) θy [ Cy + ( 4 C)], (.9) 4 C MSE( tpe) θy [ Cy + (+ 4 C)], (.0) 4 3. Propoed Etimator Motivated by Sigh et al. (04) ad Praad (989), we attempt to propoe the followig efficiet ratio ad product type epoetial etimator i predictive etimatio a, X Re ep τ κ y+ y X + ( ) κ y+ yep X (3.) ( X ) X τ ep Pe κ y+ y + X ( ) κ y+ yep X (3.) X + ( ) where κ ad κ are the cotat kow a kappa to be determied uch that the mea quare error of the etimator τ ad τ are miimum repectively. To tudy the large ample propertie of the propoed etimator τ Re ad τ Pe, we defie, y Y( + e0 ) ad X( + e ) uch that Ee ( 0) Ee ( ) 0 ad up to the firt order of approimatio, Ee ( 0 ) θ Cy, Ee ( ) θ C ad E( e0 e ) θ CC. Epreig (3.) i term of e i ad implifyig, we have τ e ee 0 e κ Y + e + (3 4 f) 8 Re 0 (3.3) Subtractig Y o both ide of (3.3), implifyig ad takig epectatio both ide, we get the bia of τ Re a, Ee ( ) ( τre) Y ( κ ) + κ Ee ( 0) Eee ( 0) Ee ( ) + (3 4 f ) 8 Puttig the value of differet epectatio, we get Y f C ( Re) ( ) (3 4 ) CC τ κ κθ + 8 (3.4)

4 America Joural of Operatioal Reearch 04, 4(): Squarig o both ide of (3.3) after ubtractig Y, epadig, implifyig up to the firt order of approimatio ad takig epectatio o both ide, we have the mea quare error of τ Re a, MSE( τre) Y Ee ( ) Ee ( ( κ ) + κ { Ee ( 0 ) Eee ( 0 ) + (5 4 f ) ) Eee ( 0) κ (3 4 f ) 8 8 which i miimum for Ee ( ) Eee ( 0) + (3 4 f ) 8 A κ Ee ( ) + Ee ( 0) Eee ( 0) + (5 4 f) 8 where Ad the miimum mea quare error of τ Re i, Ee ( ) Eee ( 0) A + (3 4 f) 8 Ee Eee f ( ) 8 Ee + ( 0) ( 0) + (5 4 ) MSE mi ( τ Re) Y A Similarly, the bia ad mea quare error of τ Pe, up to the firt order of approimatio repectively are, CC ( ) ( ) C τpe Y κ + κθ 8( f ) MSE( τpe) Y f Eee ( ( κ ) + κ{ Ee ( 0) + Eee ( 0) Ee ( ) 0) Ee ( ) κ 4( f ) 8( f ) which i miimum for where Eee ( 0) Ee ( ) + 8( f ) A κ f + Ee ( 0) + Eee ( 0) Ee ( ) 4( f ) A Eee ( 0) Ee ( ) + 8( f ) f + Ee ( 0) + Eee ( 0) Ee ( ) 4( f ) (3.5) (3.6)

5 3 Subhah Kumar Yadav et al.: Optimal Search for Efficiet Etimator of Fiite Populatio Mea Uig Auiliary Iformatio Ad the miimum mea quare error of τ Pe i, MSE mi ( τ Pe) Y A (3.7) 4. Coditio of Optimal Search ad Efficiecy Compario Uig (.5) ad (3.5), we fairly get I view of (.6) ad (3.5), we have Further uig (.9) ad (3.5), we fid { } MSE( yr) MSE( τre) Y θ Cy + C( C) { } MSE( yp or t p) MSE( τre) Y θ Cy + C(+ C) A > 0 A > 0 C MSE( tre) MSE( τre) Y θ C y ( 4 C) + 4 A > 0 Fially, i the light of (.0) ad (3.5), we happe to obtai C ( ) ( Re) A MSE tpe MSE τ Y θ Cy ( 4 C) + + > 0 4 A > 0 (4.) (4.) (4.3) (4.4) Table. The data decriptio Populatio C y C ρ C I. Steel ad Torrie (960) y: Log of leaf bur i ec. : Chlorie percetage II. Da (988) y: The umber of agricultural laborer for 96 : The umber of agricultural laborer for III. Cochra (977) y: The umber of pero per block : The umber of room per block Table. The PRE of differet etimator with repect to y Populatio PRE( y R, y ) PRE( y P or t p, y ) PRE( t Re, y ) PRE( t Pe, y ) PRE( τ Pe, y ) PRE( τ Re, y ) I II III PRE ued i the above table i computed by the followig formula. PRE (., y ) MSE( y )00/MSE(.)

6 America Joural of Operatioal Reearch 04, 4(): Above coditio are coditio of uder which the propoed etimator perform better tha the above metioed etimator of populatio mea. Remark: Similar coditio are alo draw for the etimator τ Pe to be more efficiet tha the above metioed etimator. 5. Empirical Study A empirical tudy ha bee carried out umerically to how the uefule of uggeted methodology i thi paper. To verify the theoretical fidig of the propoed etimator τ over the etimator y, y R, y P, t p, tre ad t Pe of populatio mea i predictive etimatio, we have take the followig three populatio i table. The followig table repreet the percetage relative efficiecy (PRE) of differet metioed etimator with repect to mea per uit etimator y of populatio mea Y i predictive etimatio approach. 6. Reult ad Cocluio I the paper, we have ucceeded i developig the improved efficiet etimator for poitively ad egatively correlated data for etimatig populatio mea i predictive etimatio approach. Upo drawig the obervatio from the theoretical dicuio of ectio-4 ad the reult i table-, it i evidet that the propoed etimator τ Re ad τ Pe are better tha the Srivatava (983) etimator y R, y, t P t p ad the Sigh et.al (04) etimator Re, t Pe a they have leer mea quare error tha all thee etimator. Therefore the propoed etimator τ Re ad τ Pe hould be preferred for the etimatio of populatio mea i predictive etimatio approach for poitively ad egatively correlated data repectively. REFERECES [] ahl, S. ad Tuteja, R.K. Ratio ad product type epoetial etimator, Iformatio ad optimizatio Sciece XII (I), 59-63, 99. [] Cochra, W.G. (977): Samplig Techique. 3rd ed. ew York, USA: Joh Wiley ad So. [3] Da, A. K. (988): Cotributio to the Theory of Samplig Strategie aed o Auiliary Iformatio. Ph.D. thei ubmitted to.c. K. V. Mohapur, adia, Wet egal, Idia. [4] Jeelai, M.I., Maqbool, S. ad Mir, S.A. (03): Modified Ratio Etimator of Populatio Mea Uig Liear Combiatio of Co-efficiet of Skewe ad Quartile Deviatio. Iteratioal Joural of Moder Mathematical Sciece, 6, 3, [5] Murthy, M.. (967): Samplig Theory ad Method, Statitical. Publihig Society, Calcutta. [6] Mihra S. S. ad Sigh P. K. (0): Computatioal approach to a ivetory model with ramp- type demad ad liear deterioratio, Iteratioal Joural of Operatio Reearch, p.p , Vol. 5, o. 3. [7] Mihra S. S. ad Sigh P. K. (0): Computig of total optimal cot of a EOQ model with quadratic deterioratio ad occurrece of hortage, Iteratioal joural of Maagemet Sciece ad Egieerig Maagemet, World Academic Pre, ISS , Eglad, UK 7(4): [8] Mihra S. S. ad Mihra P. P. (0): Phae Wie Supply Chai Model of EOQ with ormal Life Time for Queued Cutomer: A Computatioal Approach, America Joural of Operatio ( [9] Mihra S. S. ad Sigh P. K. (03): Partial ackloggig EOQ Model for Queued Cutomer with Power demad ad Quadratic Deterioratio: Computatioal Approach, America Joural of Operatio Reearch, Vol.3. Iue, pp Mot read paper o Scietific ad Academic Pre(USA) [0] Oyeka A.C. (0): Etimatio of populatio mea i pottratified amplig uig kow value of ome populatio parameter(). Statitic i Traitio-ew Serie 3: [] Padey H, Yadav S. K., Shukla A. K. (0): A improved geeral cla of etimator etimatig populatio mea uig auiliary iformatio. Iteratioal Joural of Statitic ad Sytem 6:7. [] Praad,. (989): Some improved ratio type etimator of populatio mea ad ratio i fiite populatio ample urvey, Commuicatio i Statitic: Theory ad Method 8, [3] Saii, M. A. (03): cla of predictive etimator i two-tage amplig whe auiliary character i etimated at SSU level. Iteratioal Joural of Pure ad Applied Mathematic, 85(), [4] Solaki R.S., Sigh H.P., Rathour A. (0): A alterative etimator for etimatig the fiite populatio mea uig auiliary iformatio i ample urvey. Iteratioal Scholarly Reearch etwork: Probability ad Statitic, 0, -4. [5] Srivatava, S.K. (983): Predictive etimatio of fiite populatio mea uig product etimator. Metrika, 30, [6] Sigh, H.P., Solaki, R.S. ad Sigh, A.K. (04): Predictive Etimatio of Fiite Populatio Mea Uig Epoetial Etimator, STATISTIKA, 94 (), [7] Steel, R.G.D., Torrie, J. H. Priciple ad Procedure of Statitic. ew York, USA: McGraw, 960. [8] Subramai, J. ad Kumarapadiya, G. (0): Etimatio of Populatio Mea Uig Co- Efficiet of Variatio ad Media of a Auiliary Variable. Iteratioal Joural of Probability ad Statitic (4): -8. [9] Yadav, S.K. (0): Efficiet etimator for populatio

7 34 Subhah Kumar Yadav et al.: Optimal Search for Efficiet Etimator of Fiite Populatio Mea Uig Auiliary Iformatio variace uig auiliary iformatio. Global Joural of Mathematical Sciece: Theory ad Practical 3: [0] Yadav, S.K. ad Kadilar, C. (03): Improved cla of ratio ad product etimator, Applied Mathematic ad Computatio, 9, [] Ya, Z. ad Tia,. (00): Ratio method to the mea etimatio uig co-efficiet of kewe of auiliary variable, ICICA, Part II, CCIS 06, pp [] Yadav S K, S S Mihra ad Alok Shukla (0): A Geeralized Cla of Regreio Type Etimator i two phae Samplig, pp. 5-8, ESMSJ, , Vol.(), 0. [3] Yadav S K, Mihra S S ad Shukla A (04): Improved ratio etimator for populatio mea baed o media uig liear combiatio of populatio mea ad media of a auiliary variable, America Joural of Operatio Reearch, Scietific ad Academic Publihig, Vol. 4, Iue, pp -7, 04.