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

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1 Rajes Sing Prayas Sarma Deparmen of Saisics Banaras Hindu Universiy aranasi-005 India Florenin Smarandace Universiy of New Mexico Gallup USA Exponenial Raio-Produc Type Esimaors Under Second Order Approximaion In Sraified Random Sampling Publised in: Rajes Sing Florenin Smarandace (Ediors) THE EFFIIENT USE OF SUPPEMENTARY INFORMATION IN FINITE POPUATION SAMPING Educaional Publiser olumbus USA 0 ISBN: pp. 8-7

2 Absrac Sing e al. (0009) inroduced a family of exponenial raio and produc ype esimaors in sraified random sampling. Under sraified random sampling wiou replacemen sceme e expressions of bias and mean square error (MSE) of Sing e al. (009) and some oer esimaors up o e firs- and second-order approximaions are derived. Also e eoreical findings are suppored by a numerical example. Keywords: Sraified Random Sampling populaion mean sudy variable auxiliary variable exponenial raio ype esimaor exponenial produc esimaor Bias and MSE.. INTRODUTION In survey sampling i is well esablised a e use of auxiliary informaion resuls in subsanial gain in efficiency over e esimaors wic do no use suc informaion. However in planning surveys e sraified sampling as ofen proved needful in improving e precision of esimaes over simple random sampling. Assume a e populaion U consis of sraa as U=U U U. Here e size of e sraum U is N and e size of simple random sample in sraum U is n were = ---. Wen e populaion mean of e auxiliary variable X is nown Sing e al. (009) suggesed a combined exponenial raio-ype esimaor for esimaing e populaion mean of e sudy variable Y : 8

3 X x s S yexp (.) X x s were n n y yi x x i i n i n ys w y xs w x and X w X. Te exponenial produc-ype esimaor under sraified random sampling is given by X x s S yexp (.) x s X Following Srivasava (967) an esimaor s in sraified random sampling is defined as : X x s S yexp (.) x s X were α is a consan suiably cosen by minimizing MSE of convenional exponenial raio-ype esimaor wereas for exponenial produc ype esimaor.. For α= is same as α = - i becomes convenional Sing e al. (008) inroduced an esimaor wic is linear combinaion of exponenial raioype and exponenial produc-ype esimaor for esimaing e populaion mean of e sudy variable Y in simple random sampling. Adaping Sing e al. (008) esimaor in sraified random sampling we propose an esimaor s as : X x s X x s S yexp ( )exp (.) x s X x s X 9

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

5 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 r y Y x X N s r were y Y x X rs () N i () () W YX () () W 0 Y X () 0() W Y () 0() 0 W X () () () 0() 0() W YX

6 () 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 ). Firs Order Biases and Mean Squared Errors under sraified random sampling Te expressions for biases and MSEs of e esimaors 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 (.) Bias ( ) Y (.5)

7 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 esimaor. Te expression for e bias and MSE of s o e firs 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 esimaor. We observe a for e opimum cases e biases of e esimaors and S are differen bu e MSE of and S are same. I is also observed a e MSE s of e esimaors and S are always less an e MSE s of e esimaors S and S. Tis promped us o sudy e esimaors and S under second order approximaion. 5. Second Order Biases and Mean Squared Errors in sraified random sampling Expressing esimaor i s(i=) in erms of e i s (i=0) we ge s Y e e exp 0 e Or

8 e s Y Ye 0 e0e e e0e e e0e e (5.) Taing expecaions we ge e bias of e esimaor approximaion as s up o e second order of 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 ( s ) Y Ee 0 e e0e e0 e e0 e e e0e e0e e 8 9 Or (5.) MSE Y s Similarly we ge e biases and MSE s of e esimaors S and S up o second order of approximaion respecively as (5.) Y 5 5 Bias ( s ) 0 0 (5.5) 0 9 MSE Y ) (5.6) S

9 Bias ( ) Y (5.7) MSE Y (5.8) Bias ( S ) E( S Y) 6 Y (5.9) MSE S Y (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 esimaor s is obained by using numerical ecniques. 6. Numerical Illusraion 5

10 For e one naural populaion daa we sall calculae e bias and e mean square error of e esimaor and compare Bias and MSE for e firs and second order of approximaion. Daa Se- To illusrae e performance of above esimaors we ave considered e naural daa given in Sing and audary (986 p.6). Te daa were colleced in a pilo survey for esimaing e exen of culivaion and producion of fres fruis in ree disrics of Uar- Prades in e year Table 6.: Bias and MSE of esimaors Esimaor Bias MSE Firs order Second order Firs order Second order s s s s ONUSION In e Table 6. e bias and MSE of e esimaors S S and S are wrien under firs order and second order of approximaion. Te esimaor S is exponenial produc-ype esimaor and i is considered in case of negaive correlaion. So e bias and mean squared error for is esimaor is more an e oer esimaors considered ere. For e classical exponenial raio-ype esimaor i is observed a e biases and e mean squared errors increased for second order. Te esimaor and S ave e same mean squared error for e firs order bu e mean squared error of is less an S for e second order. So on 6

11 e basis of e given daa se we conclude a e esimaor is bes followed by e esimaor S among e esimaors considered ere. REFERENES Koyuncu N. and Kadilar. (009) : Family of esimaors of populaion mean using wo auxiliary variables in sraified random sampling. ommun. in Sais. Teor. and Me Koyuncu N. and Kadilar. (00) : On e family of esimaors of populaion mean in sraified random sampling. Pa. Jour. Sa. 6()7-. Sing D. and udary F.S. (986): Teory and analysis of sample survey designs. Wiley Easern imied New Deli. Sing R. auan P. and Sawan N.(008): On linear combinaion of Raio-produc ype exponenial esimaor for esimaing finie populaion mean. Saisics in Transiion9()05-5. Sing R. Kumar M. audary M. K. Kadilar. (009) : Improved Exponenial Esimaor in Sraified Random Sampling. Pa. J. Sa. Oper. Res. 5() pp Sing R. and Smarandace F. (0): On improvemen in esimaing populaion parameer(s) using auxiliary informaion. Educaional Publising & Journal of Maer Regulariy (Beijing) pg 5-. 7