Exponential Ratio-Product Type Estimators Under Second Order Approximation In Stratified Random Sampling

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1 Exponenial Raio-Produc Type Eimaors Under Second Order Approximaion In Sraified Random Sampling Rajes Sing Prayas Sarma and Florenin Smarandace Deparmen of Saiics Banaras Hindu Universiy aranasi-005 India air of Deparmen of Maemaics Universiy of New Mexico Gallup USA Abrac Sing e al. (0009) inroduced a family of exponenial raio and produc ype eimaors in raified random sampling. Under raified random sampling wiou replacemen sceme e expressions of bias and mean square error (MSE) of Sing e al. (009) and some oer eimaors up o e fir- and second-order approximaions are derived. Also e eoreical findings are suppored by a numerical example. Keywords: Sraified Random Sampling populaion mean udy variable auxiliary variable exponenial raio ype eimaor exponenial produc eimaor Bias and MSE.. INTRODUTION In survey sampling i is well eablised a e use of auxiliary informaion resuls in subanial gain in efficiency over e eimaors wic do no use suc informaion. However in planning surveys e raified sampling as ofen proved needful in improving e precision of eimaes over simple random sampling. Assume a e populaion U consi of raa as U=U U U. Here e size of e raum U is N and e size of simple random sample in raum U is n were = ---. Wen e populaion mean of e auxiliary variable X is nown Sing e al. (009) suggeed a combined exponenial raio-ype eimaor for eimaing e populaion mean of e udy variable Y :

2 X x S yexp (.) X x were n n y yi x x i i n i n y w y x w x and X w X. Te exponenial produc-ype eimaor under raified random sampling is given by X x S yexp (.) x X Following Srivaava (967) an eimaor s in raified random sampling is defined as : X x S yexp (.) x X were α is a conan suiably cosen by minimizing MSE of. For α= is same as convenional exponenial raio-ype eimaor wereas for α = - i becomes convenional exponenial produc ype eimaor. Sing e al. (008) inroduced an eimaor wic is linear combinaion of exponenial raio-ype and exponenial produc-ype eimaor for eimaing e populaion mean of e udy variable Y in simple random sampling. Adaping Sing e al. (008) eimaor in raified random sampling we propose an eimaor s as : X x X x S yexp ( )exp (.) x X x X

3 were is e conan and suiably cosen by minimizing mean square error of e eimaor S. I is observed a e eimaors considered ere are equally efficien wen erms up o fir order of approximaion are aen. Hossain e al. (006) and Sing and Smarandace (0) udied some eimaors in SRSWOR under second order approximaion. Koyuncu and Kadilar (009 00) ) ave udied some eimaors in raified random sampling under second order approximaion. To ave more clear picure abou e be eimaor in is udy we ave derived e expressions of MSE s of e eimaors considered in is paper raified random sampling. up o second order of approximaion in. Noaions used e us define e 0 y y and e y x x x suc a rs W rs E r x X y Y s To obain e bias and MSE of e proposed eimaors we use e following noaions in e re of e aricle: were and are e sample and populaion means of e udy variable in e raum respecively. Similar expressions for X and Z can also be defined. Also we ave

4 were f n n f N N w. n Some addiional noaions for second order approximaion: rs W rs r Y X s E s y Y x X r N s r were y Y x X rs() N i () () W YX () () W 0 Y X () 0() W Y

5 () 0() 0 W X () () () 0() 0() W YX () 0() () 0() 0 W X () () () 0() 0() () W Y X were () (N n )(N n ) n (N )(N ) () (N n )(N )N 6n (N n ) n (N )(N )(N ) () (N n )N (N n )(n ). n (N )(N )(N ). Fir Order Biases and Mean Squared Errors under raified random sampling Te expressions for biases and MSEs of e eimaors S S and respecively are : Bias ( S ) Y 0 8 (.) MSE ( S ) Y 0 0 (.) Bias ( S ) Y 0 8 (.) MSE ( S ) Y 0 0 (.)

6 Bias ( ) Y (.5) MSE ( ) Y 0 0 (.6) By minimizing MSE (s) e opimum value of is obained as o 0. By puing is opimum value of in equaion (.5) and (.6) we ge e minimum value for bias and MSE of e eimaor. Te expression for e bias and MSE of s o e fir order of approximaion are given respecively as Bias ( s) Y 0 ( ) 0 (.7) 8 8 MSE ( S ) Y 0 0 (.8) By minimizing MSE ( S) e opimum value of is obained as o 0. By puing is opimum value of in equaion (.7) and (.8) we ge e minimum value for bias and MSE of e eimaor. We observe a for e opimum cases e biases of e eimaors and S are differen bu e MSE of and S are same. I is also observed a e MSE s of e eimaors and S are always less an e MSE s of e eimaors S and S. Tis promped us o udy e eimaors and S under second order approximaion. 5. Second Order Biases and Mean Squared Errors in raified random sampling Expressing eimaor i s(i=) in erms of e i s (i=0) we ge

7 s Y e e exp 0 e Or e s Y Ye 0 e0e e e0e e e0e e (5.) as Taing expecaions we ge e bias of e eimaor s up o e second order of approximaion Bias Y ) (s (5.) Squaring equaion (5.) and aing expecaions and using lemmas we ge MSE of order of approximaion as s up o second MSE( Or e S ) E Ye0 e e0e e0e e 8 MSE ( Or s ) Y Ee 0 e e0e e0 e e0 e e e0e e0e e 8 9 (5.) MSE Y s Similarly we ge e biases and MSE s of e eimaors S and S approximaion respecively as 5 5 (5.) up o second order of

8 Y 5 5 Bias ( s ) 0 0 (5.5) 0 9 MSE Y ) (5.6) S Bias ( ) Y (5.7) MSE Y (5.8) Bias ( S ) E( S Y) 6 Y (5.9) MSE S Y

9 5 (5.0) Te opimum value of we ge by minimizing MSE. Bu eoreically e deerminaion of e opimum value of is very difficul we ave calculaed e opimum value by using numerical ecniques. Similarly e opimum value of wic minimizes e MSE of e eimaor s is obained by using numerical ecniques. 6. Numerical Illuraion For e one naural populaion daa we sall calculae e bias and e mean square error of e eimaor and compare Bias and MSE for e fir and second order of approximaion. Daa Se- To illurae e performance of above eimaors we ave considered e naural daa given in Sing and audary (986 p.6). Te daa were colleced in a pilo survey for eimaing e exen of culivaion and producion of fres fruis in ree dirics of Uar- Prades in e year Table 6.: Bias and MSE of eimaors Eimaor Bias MSE Fir order Second order Fir order Second order s s s s

10 7. ONUSION In e Table 6. e bias and MSE of e eimaors S S and S are wrien under fir order and second order of approximaion. Te eimaor S is exponenial produc-ype eimaor and i is considered in case of negaive correlaion. So e bias and mean squared error for is eimaor is more an e oer eimaors considered ere. For e classical exponenial raio-ype eimaor i is observed a e biases and e mean squared errors increased for second order. Te eimaor and S ave e same mean squared error for e fir order bu e mean squared error of is less an S for e second order. So on e basis of e given daa se we conclude a e eimaor is be followed by e eimaor S among e eimaors considered ere. REFERENES Koyuncu N. and Kadilar. (009) : Family of eimaors of populaion mean using wo auxiliary variables in raified random sampling. ommun. in Sai. Teor. and Me Koyuncu N. and Kadilar. (00) : On e family of eimaors of populaion mean in raified random sampling. Pa. Jour. Sa. 6()7-. Sing D. and udary F.S. (986): Teory and analysis of sample survey designs. Wiley Eaern imied New Deli. Sing R. auan P. and Sawan N.(008): On linear combinaion of Raio-produc ype exponenial eimaor for eimaing finie populaion mean. Saiics in Transiion9()05-5. Sing R. Kumar M. audary M. K. Kadilar. (009) : Improved Exponenial Eimaor in Sraified Random Sampling. Pa. J. Sa. Oper. Res. 5() pp 67-8.

11 Sing R. and Smarandace F. (0): On improvemen in eimaing populaion parameer(s) using auxiliary informaion. Educaional Publising & Journal of Maer Regulariy (Beijing) pg 5-.