# Some Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation

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3 P p p P yexp ( ) exp P p P p t (.) where is the costat ad suitably chose by miimizig mea square error of the estimator t. For = estimator t reduces to expoetial ratio-type estimator ad for =0, it reduces to expoetial product-type estimator.. Notatios used y Y p P et us defie, e 0 ad e, the E(e0 ) E(e) =0. Y P For obtaiig the bias ad the followig lemmas will be used: emma. (i) N V(e0 ) E{(e0 ) } 0 0 N (vii) E {(e )} (ii) N V(e) E{(e) } 0 0 N N (iii) OV(e0, e) E{(e0e)} N emma. (iv) (N ) (N ) E{(e e0 )} (N ) (N ) (v) (N ) (N ) E{(e )} 0 0 (N ) (N ) emma. (vi) E(e e 0 ) 0 (N ))(N N 6N 6 (N )(N )(N ) (viii) E(e ) e0 ) ( ) Where (N )(N N 6N 6 ) ad (N )(N )(N ) N(N )(N )( ) (N )(N )(N ) N r s ( i - P) (Yi - Y) ad rs =. r s N P Y i 60

5 5. Secod Order Biases ad Mea Squared Errors Expressig estimator t i s (i=,,,) i terms of e s (i=0,), we get t Y e e e exp 0 Or e t Y Y e e e e0 e0e e0e e0e (5.) Takig expectatios, we get the bias of the estimator t up to the secod order of approximatio as Bias (t ) E(t Y) Y ( (5.) 0 ) Similarly, we get the biases of the estimators t, t ad t up to secod order of approximatio, respectively as Bias Y 5 5 ) E(t Y) (t 9 (0 0 ) (5.) Bias Y) Y 8 8 (t ) E(t (5.) Bias (t ) E(t Y) Y (5.5) 6

6 Squarig equatio (5.) ad tha takig expectatios ad usig lemmas, we get of t up to secod order of approximatio as Or e (5.6) (t) E Ye0 e e0e e0e e (t) Y Ee 0 e e 0e e 0 e e 0 e e e 0e e 0e e 8 9 Or t Y ( ( ( 0 0 )) 0 0 (5.7) 8 Similarly, we get the s of the estimators t, t ad t up to secod order of approximatio, respectively as t E(t Y) Y t E(t Y) Y ( ) (00 ) 0 (5.8) ) (5.9) t E(t Y) Y 0 0 6

7 (5.0) The optimum value of we get by miimizig t. But theoretically the determiatio of the optimum value of is very difficult, we have calculated the optimum value by usig umerical techiques. Similarly, the optimum value of which miimizes the of the estimator t is obtaied by usig umerical techiques. 6. Numerical Illustratio The various results obtaied i the previous sectios are ow examied with the help of the followig data : Source of the Data The data for the empirical aalysis are take from Sukhatme ad Sukhatme (970), p.56 ad the followig values are obtaied ad surmised i the Table 6.. Table 6.: Biases ad s of estimators Estimator Bias First Order Secod order First order Secod Order t t t t

9 Srivastava, S.K., 967, A estimator usig auxiliary iformatio i sample surveys. al. Stat. Ass. Bull. 5:7-. Sukhatme, P.V. ad Sukhatme, B.V., 970, Samplig theory of surveys with applicatios. Iowa State Uiversity Press, Ames, U.S.A. Upadhyaya,. N. ad Sigh, H. P., 999, Use of trasformed auxiliary variable i estimatig the fiite populatio mea. Biom. Jour.,,

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