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

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1 ; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN )]; ISSN X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial Ratio-Product Type Estimators usig iformatio o Auxiliary Attributes uder Secod Order Approximatio Prayas Sharma, Hemat K. Verma, Aamir Saaullah ad Rajesh Sigh Departmet of Statistics, Baaras Hidu Uiversity,Varaasi, Idia Natioal ollege of Busiess Admiistratio ad Ecoomics, ahore,pakista orrespodig author ABSTRAT Bahl ad Tuteja (99) itroduced a expoetial ratio type ad expoetial product-type estimators for estimatig populatio mea Y. Sigh et al. (007) suggested improved estimator usig Bahl ad Tuteja (99) estimator. Most of them discussed these estimators alog with their first order biases ad mea square error s ( s). I this paper, we have tried to work out the secod order biases ad mea square errors of some estimators usig iformatio o auxiliary attributes based o simple radom samplig. Fially, we have compared the performace of the estimators with help of some umerical illustratio. Keywords: Simple Radom Samplig, populatio mea, study variable, auxiliary attributes, expoetial ratio type estimator, expoetial product estimator, attributes, Bias ad. Mathematics Subject lassificatio: 6D05 Joural of Ecoomic iterature (JE) lassificatio Number : 8. INTRODUTION I survey samplig use of auxiliary iformatio ca icrease the precisio of a estimator whe study variable y is highly correlated with the auxiliary variable x. May authors suggested estimators usig some kow populatio parameters of a auxiliary variable. Upadhyaya ad Sigh (999), Sigh ad Tailor (00), Khoshevisa et al. (007), Sigh et al. (007), Sigh et al. (008) ad Sigh ad Kumar (0) suggested estimators i simple radom samplig. But i several practical situatios, istead of existece of auxiliary variable there exists some auxiliary attributes which are highly correlated with study variable y. For example: (i) Sex is a good auxiliary attribute while dealig with height. (ii) Milk produced ad particular breed of cow. (iii) Yield of wheat crop ad a particular variety of wheat (Shabbir ad Gupta (007))

2 I such situatios, takig the advatage of poit bi-serial correlatio betwee the study variable ad the auxiliary attribute, the estimators of parameters of iterest ca be costructed by usig prior kowledge of the parameter of auxiliary attributes.shabbir ad Gupta (007), Sigh et al. (007), Sigh et al. (00), Abd-Elfattah et al.(00), Sigh ad Solaki (0) ad Malik ad Sigh (0) have cosidered the problem of estimatig populatio mea usig poit bi-serial correlatio betwee study variable ad auxiliary attribute. osider a sample of size draw by simple radom samplig without replacemet (SRSWOR) from a populatio of size N. let y i ad xi deote the observatio o variable y ad x respectively for the i th uit (i=,, N). We ote that N i =, if i th uit possesses attribute ad i =0 otherwise. et A i, a i deotes the total umber of uits i the populatio ad sample respectively, i i possessig attribute. et A P ad N sample respectively possessig attribute x. a p deotes the proportio of uits i the populatio ad. Some Estimators i Simple Radom Samplig For estimatig the populatio mea Y of Y, the, Sigh et al. (008) estimator t is give by where y y i i P p yexp P p t (.) Sigh et al. (008) product type estimator t is give by p P yexp P p t (.) Followig Srivastava (967) a estimator t is defied as P p y exp P p t (.) where is a costat suitably chose by miimizig of the estimator t. For =, t is same as covetioal expoetial ratio-type estimator, whereas for = -, it becomes covetioal expoetial product-type estimator. For estimatig the populatio mea Y, Sigh et al. (007) defied a estimator t, as 59

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

4 Proof of these lemma are straight forward by usig SRSWOR ( see Sukhatme ad Sukhatme (970)).. First Order Biases ad Mea Squared Errors The bias of the estimators t, t ad t ad t are respectively writte as Bias (t) Y 0 8 (.) Bias (t ) Y 0 (.) Bias (t ) Y (.) Bias(t ) Y 0 ( ) 0 (.) 8 8 of the estimators t, t, t ad t are respectively give by, (t) Y 0 0 (.5) (t ) Y 0 0 (.6) (t ) Y 0 0 (.7) (t ) Y 0 0 (.8) By miimizig (t ), the optimum value of is obtaied as o 0. By puttig this optimum value of i equatio (.7) we get the miimum value for bias ad of the estimator t. By miimizig (t ), the optimum value of is obtaied as o. By puttig this 0 optimum value of i equatio (.8), we get the miimum value for bias ad of the estimator t. We observe that for the optimum cases the biases of the estimators t ad t are differet but the of t ad t are same. It is also observed that the s of the estimators t ad t are always less tha the s of the estimators t ad t. This prompted us to study the estimators t ad t uder secod order approximatio. 6

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

8 7. ONUSION I the Table 6. the biases ad s of the estimators t, t, t ad t are writte uder first order ad secod order of approximatios. The estimator t is expoetial product estimator for auxiliary attributes ad it is cosidered i case of egative correlatio. So the bias ad mea squared error for this estimator is more tha the other estimators cosidered here. For the classical expoetial ratio-type estimator i case of auxiliary attributes, it is observed that the biases ad the mea squared errors icrease for secod order. The estimator t ad t have the same mea squared error for the first order but the mea squared error of t is less tha t for the secod order. So, o the basis of the give data set, we coclude that the estimator t is best followed by the estimator t i the presece of auxiliary attributes, amog the estimators cosidered here. 8. REFERENES Abd-Elfattah, A.M., Sherpey E.A., Mohamed S.M., Abdou O.F.,00, Improvemet estimatig the populatio mea i simple radom samplig usig iformatio o auxiliary attribute. Applied mathematics ad computatio. Bahl, S. ad Tuteja, R.K., 99, Ratio ad product type expoetial estimator. Iformatio ad Optimizatio Sciece XIII Khoshevisa, M., Sigh, R., hauha, P., Sawa, N., ad Smaradache, F., 007, A geeral family of estimators for estimatig populatio mea usig kow value of some populatio parameter(s), Far East Joural of Theoretical Statistics 8 9. Malik, S. ad Sigh, R., 0, A family of estimators of populatio mea usig iformatio o poit bi-serial ad phi correlatio coefficiet. IJSE Shabbir J.,Gupta S.,007, O estimatig the fiite populatio mea with kow populatio proportio of a auxiliary variable. Pak. J. Statist.,(),-9. Sigh, H.P. ad Solaki, R. S., 0, Improved estimatio of populatio mea i simple radom samplig usig iformatio o auxiliary attribute. Appl. Math. omput., 8, Sigh, R., auha, P., Sawa, N., ad Smaradache, F., 007, Auxiliary iformatio ad a priori values i costructio of improved estimators. Reaissace High Press. Sigh, R., Kumar, M. ad Smaradache, F., 00, Ratio estimators i simple radom samplig whe study variable is a attribute. World Applied Scieces Joural (5) pp Sigh, R., hauha, P. ad Sawa, N., 008, O liear combiatio of Ratio-product type expoetial estimator for estimatig fiite populatio mea. Statistics i Trasitio,9(),05-5. Sigh, R., Kumar, M. ad Smaradache, F., 008, Almost ubiased estimator for estimatigpopulatio mea usig kow value of some populatio parameter(s). Pak. J. Stat. Oper. Res., () Sigh, R. ad Kumar, M., 0, A ote o trasformatios o auxiliary variable i survey samplig. MASA, 6:,

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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