A Family of Unbiased Estimators of Population Mean Using an Auxiliary Variable

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1 Advaces i Computatioal Scieces ad Techolog ISSN Volume 10, Number 1 (017 pp Research Idia Publicatios A Famil of Ubiased Estimators of Populatio Mea Usig a Auiliar Variable Shashi Bhusha ad Pravee Kumar Misra Departmet of Mathematics ad Statistics, Dr. Shakutala Misra Natioal Rehabilitatio Uiversit, Idia Abstract I the preset stud, we have proposed a famil of estimators give b Khoshevisa et al (007 for populatio mea of the stud variable usig iformatio o a auiliar variable uder the jack kife techique. Its ubiasedess ad mea squared error (MSE are derived. The compariso of the proposed ubiased samplig strateg with the usual ubiased estimator ad other estimators available i the literature are carried out. The results obtaied are illustrated umericall usig a empirical sample cosidered i the literature. Kewords: Famil of estimators, auiliar iformatio, bias, MSE. 1. INTRODUCTION The use of auiliar iformatio to icrease the precisio of estimators has bee discussed etesivel b various authors. The ratio estimator amog the most commol used estimator of the populatio mea or total of some variable of iterest of a fiite populatio with the help of a auiliar variable whe the correlatio coefficiet betwee the two variables is positive. I case of egative correlatio, product estimator is used. These estimators are more efficiet i.e. has smaller variace tha the usual estimator of the populatio mea based o the sample mea of a simple radom sample. I this paper, a class of estimators give b Khoshevisa et al (007 is recosidered b usig the geeralized jack kife statistic of Gra ad Schuca i sectio. I a fiite populatio, let deote the variable whose populatio mea Y is to be estimated b usig iformatio of the auiliar variable. Assumig that the

2 130 Shashi Bhusha ad Pravee Kumar Misra populatio mea ad of the auiliar variable are kow the followig class of estimators of the populatio mea were give b Khoshevisa et al (007 a b ( a b (1 ( a b g where deotes the sample mea of the variable of iterest ;, respectivel deote the sample ad the populatio meas of the variables ; a ad b are real umbers or the fuctios of the kow parameters of the auiliar variable. 1.1 Defiitios ad results Assumig that the populatio mea of the auiliar variables is kow. Let, respectivel deote the sample meas of the variables, based o a simple radom sample without replacemet (SRSWOR of size draw from the populatio. Defie: Y 0, e e f f 1, The E( e0 E( e1 0, E( e0 C, E( e1 C, Y f E( e e 0 1 CC where N N S S 1 f, C, C, S N 1 Yi Y, N Y i1 S S N Y Y N 1 1 i i S i 1 S Also, deote the correlatio coefficiet betwee Y ad,respectivel ad C, C ad deote the coefficiet of variatio of Y, ad respectivel. (1 1. Properties of the estimator Puttig these values, we have gg1 Y 1 e0 ge1 e1 ge1e 0... where a a b ( Takig epectatio of (, we have gg1 E( = YE 1 e0 ge1 e1 ge1e 0... Ad the bias of the estimator is give b f g( g 1 B ( Y C gcc (3 (4 Ad its mea square error is give b f MSE( E( Y Y C gc gcc (5

3 A Famil of Ubiased Estimators of Populatio Mea Usig a Auiliar Variable 131 The miimum MSE of the class of estimators is obtaied b miimizig with respect K C to we get the optimum value opt where K g C ( f MSEmi Y 1 C (6 The MSE ( mi is same as that of the approimate variace of the usual liear regressio estimator.. THE PROPOSED JACK-KNIFE ESTIMATOR j Let a simple radom sample of size =m is draw without replacemet from the populatio of size N. This sample of size =m is the split up at radom i to two sub samples each of size m i a b ( a b (1 ( a b g g a b, i=1, (8 ( ai b (1 ( a b (7 Let us defie j R 1 R where 1 ad R B ( B ( B aalog to (4, we have f g( g 1 B ( Y C gcc = B1(sa (9 Similarl, it ca be show that f g( g 1 B ( Y C i gcc m, i=1, f g( g 1 B Y C g CC m Thus, ( (10 E( RE( Now, takig epectatio of j, we have E ( j Y 1 R thereb showig that is a ubiased estimator of populatio mea Y. j.1 Mea square error of j R MSE( j E( j Y E Y 1 R = E( Y R E( Y R( Y ( Y (1 R (11

4 13 Shashi Bhusha ad Pravee Kumar Misra MSE( E( Y f Y C gc gcc (1 1 1 ( ( Y Y E( Y E Y E 1 ( ( ( ( E Y E Y E Y Y f E Y Y C gc gc C i m (1 (1 ( ( E( Y ( Y Y ( e0 ge1 ( e0 ge1 m ( ; 1, i 1 (13 Y E( e e g( e e e e g e e (1 ( (1 ( (1 ( (1 ( Y = C gc gcc N (14 Puttig these values, we have E( Y = 1 N m Y C gc gcc 4 mn N f = Y C gc gcc 1 E( Y ( Y Y ( Y 1 ( ( ( ( 1 E Y Y E Y Y ( i ( i E( Y ( Y Y ( e0 ge1 ( e0 ge1 i (15 Y E e e g( e e e e g e e ( i ( i ( i ( i f E Y Y Y C gc gc C i ( ( ; 1, i Puttig all the results i (11, we get f MSE( j Y C gc gcc (16 which is same as the MSE of. 3. COMPARISON OF THE ESTIMATORS We cosider the followig kow estimators belogig to the cosidered famil of estimators 1. For 0, a 0, b 0 ad g 0 i the proposed class of estimators SRS mea.. The ratio estimator for 1, a 1, b 0 ad g 1 YR 3. Product estimator for 1, a 1, b 0 ad g 1 YP

5 A Famil of Ubiased Estimators of Populatio Mea Usig a Auiliar Variable Sisodia ad dwivedi (1981 estimator for 1, a 1, b C ad g 1, C sd C 5. Pade ad Dube (1988 estimator for 1, a 1, b C ad g 1 C pd C 6. Upadhaa ad Sigh (1999 estimator for 1, a (, b C ad g 1 ( C u s1 ( C 7. Upadhaa ad Sigh (1999 estimator for 1, a C, b ( ad g 1 C ( u s C ( 8. G.N.sigh(003 estimator for 1, a 1, b ad g 1 GN1 9. G.N.sigh (003 estimator for 1, a 1(, b ad g 1 1( GN 1( 10. G.N.sigh (003 estimator for 1, a (, b ad g 1 ( GN 3 ( 11. Sigh,Tailor (003 estimator for 1, a 1, b ad g 1 TL1 1. Sigh,Tailor (003 estimator for 1, a 1, b ad g 1 TL 13. Sigh,et.al.(004 estimator for 1, a 1, b ( ad g 1 ( si gh1 ( 14. Sigh, et.al. (004 estimator for 1, a 1, b ( ad g 1 ( ( si gh The MSE of the above estimators up to the first order of approimatio is give b Var( = f f C, MSE( R Y ( C C CC

6 134 Shashi Bhusha ad Pravee Kumar Misra f f MSE( P Y ( C C CC, MSE( sd Y ( C 1C 1CC f f MSE( pd Y ( C 1C 1CC, MSE( u s1 Y ( C C CC f f MSE( u s Y ( C 3 C 3CC, MSE( GN1 Y ( C 4 C 4CC f f MSE( GN Y ( C 5 C 5CC, MSE( GN 3 Y ( C 6 C 6CC f f MSE( TL 1 Y ( C 7 C 7CC, MSE( TL Y ( C 7 C 7CC f MSE( si gh1 Y ( C 8 C 8CC, f MSE( si gh Y ( C 8 C 8CC where 1 C 1( 5 1(,, 6 ( ( C ( (, 3, 7 C C C, 4, 8 C ( O comparig the efficiec of the proposed jack-kife estimator with above metioed estimators, we have MSE( R MSE( j 0, MSE( P MSE( j 0, MSE( sd MSE( j 0 MSE( pd MSE( j 0, MSE( s1 MSE( j 0 u, u s MSE( MSE( j 0 MSE( GN 1 MSE( j 0, MSE( GN MSE( j 0, MSE( GN 3 MSE( j 0 MSE( TL 1 MSE( j 0, MSE( TL MSE( j 0, MSE( si gh1 MSE( j 0 MSE( si gh MSE( j 0 Thus, the proposed jack-kife estimators j is ubiased ad has got lesser mea square error as compared with the above estimators. 4. NUMERICAL ILLUSTRATION We cosider the data used b Pade ad Dube(1988 to demostrate what we have discussed i the above sectios. The followig values were obtaied usig the whole data set: C , C 0.16 N=0, =8Y =19.55, =18.8, , ( , ( , Usig the above results we calculated the percet relative efficiec (PRE of differet estimators i Table1,

7 A Famil of Ubiased Estimators of Populatio Mea Usig a Auiliar Variable 135 Table1: PRE of estimators uder stud based o populatio data Estimator PRE 100 R P 3.91 sd pd u s u s GN GN GN 3.17 TL TL 7.1 si gh1 si gh kosh j CONCLUSION It ma be easil observed from the precedig sectios, we observe that the proposed jack-kife estimator is preferable over all the cosidered estimators as it is ubiased ad efficiet. REFERENCES [1] Bhusha, S. (013. Improved Samplig Strategies i Fiite Populatio. Scholars Press, Germa. [] Bhusha S. (01. Some Efficiet Samplig Strategies based o Ratio Tpe Estimator, Electroic Joural of Applied Statistical Aalsis, 5(1, [3] Bhusha S., Gupta R. ad Pade S. K. (015. Some log-tpe classes of estimators usig auiliar iformatio, Iteratioal Joural of Agricultural ad Statistical Scieces, 11(, [4] Bhusha S. ad Katara, S. (010. O Classes of Ubiased Samplig Strategies, Joural of Reliabilit ad Statistical Studies, 3(,

8 136 Shashi Bhusha ad Pravee Kumar Misra [5] Bhusha, S. ad Kumar S. (016. Recet advaces i Applied Statistics ad its applicatios. LAP Publishig. [6] Bhusha S., Pade A. ad Sigh R.K. (009 Improved Classes of Regressio Tpe Estimators ; Iteratioal Joural of Agricultural ad Statistical Scieces (ISSN: , 5(1, [7] Bhusha S. ad Pade A. (010. Modified Samplig Strategies usig Correlatio Coefficiet for Estimatig Populatio Mea, Joural of Statistical Research of Ira, 7(, [8] Bhusha S., Sigh, R. K. ad Katara, S. (009. Improved Estimatio uder Midzuo Lahiri Se-tpe Samplig Scheme, Joural of Reliabilit ad Statistical Studies, (, [9] Bhusha S., Masalda R. N. ad Gupta P. K. (011. Improved Samplig Strategies based o Modified Ratio Estimator, Iteratioal Joural of Agricultural ad Statistical Scieces, 7(1, [10] M. Khoshevisa, Rajesh Sigh, Pakaj Chauha, Nirmala Sawa ad Floreti Smaradache(007 A Geeral Famil Of Estimators For Estimatig Populatio Mea Usig Kow Value Of Some Populatio Parameter(S, Far East Joural of Theoretical Statistics, Volume (, [11] Pade, B.N. ad Dube, Vas (1988: Modified product estimator usig coefficiet of variatio of auiliar variate, Assam Statistical Rev., (, [1] Redd, V.N. (1973: O ratio ad product methods of estimatio. Sakha, B, 35(3, [13] Sigh, G.N. (003: O the improvemet of product method of estimatio i sample surves. Jour. Id. Soc. Agri. Statistics, 56(3, [14] Sigh H.P. Ad Tailor, R. (003: Use of kow correlatio coefficiet i estimatig the fiite populatio mea. Statistics i Trasitio, 6,4, [15] Sigh H.P.,Tailor, R. ad Kakara, M.S. (004: A estimator of Populatio mea usig power trasformatio. J.I.S.A.S., 58(, [16] Sigh, J. Pade, B.N. ad Hirao, K. (1973: O the utilizatio of a kow coefficiet of kurtosis i the estimatio procedure of variace. A. Ist. Stat. Math., 5, [17] Sisodia, B.V.S. Ad Dwivedi, V.K. (1981: A modified ratio estimator usig coefficiet of variatio of auiliar variable. Jour. Id. Soc. Agril. Statist., 33,, [18] Searls, D.T. (1964: The utilizatio of kow coefficiet of variatio i the estimatio procedure. Joural of America Statistical Associatio, 59,

9 A Famil of Ubiased Estimators of Populatio Mea Usig a Auiliar Variable 137 [19] Searls, D.T. ad Itarapaich, P. (1990: A ote o a estimator for the variace that utilizes the kurtosis. The America Statisticia, 44, 4, [0] Se, A.R. (1978: Estimatio of the populatio mea whe the coefficiet of variatio is kow. Commu. Statist., Theor Meth. A (7, [1] Srivastava, S.K. (1967: A estimator usig auiliar iformatio. Calcutta Statist. Assoc. Bull., 16, [] Upadhaa, L.N. ad Sigh, H.P. (1999: Use of trasformed auiliar variable i estimatig the fiite populatio mea. Biometrical Joural, 41, 5, [3] Walsh, J.E. (1970: Geeralizatio of ratio estimator for populatio total. Sakha, A, 3,

10 138 Shashi Bhusha ad Pravee Kumar Misra